# Exponential and logarithm of multivector in low dimensional ($n=p+q<3$)
Clifford algebras

Adolfas Dargys<EMAIL_ADDRESS>Center for Physical Sciences and
Technology, Semiconductor Physics Institute, Saulėtekio 3, LT-10257 Vilnius,
Lithuania  Artūras Acus<EMAIL_ADDRESS>Institute of Theoretical
Physics and Astronomy, Vilnius University, Saulėtekio 3, LT-10257 Vilnius,
Lithuania

###### Abstract

Closed form expressions for a multivector exponential and logarithm are
presented in real Clifford geometric algebras $\mathit{Cl}_{p,q}$ when
$n=p+q=1$ (complex and hyperbolic numbers) and $n=2$ (Hamilton, split and
conectorine quaternions). Starting from $\mathit{Cl}_{0,1}$ and
$\mathit{Cl}_{1,0}$ algebras wherein square of a basis vector is either $-1$
or $+1$, we have generalized exponential and logarithm formulas to 2D
quaternionic algebras, $\mathit{Cl}_{0,2}$, $\mathit{Cl}_{1,1}$, and
$\mathit{Cl}_{2,0}$. The sectors in the multivector coefficient space where 2D
logarithm exists are found. They are related with a square root of the
multivector.

###### keywords:

Clifford (geometric) algebra, exponential and logarithm of Clifford numbers,
quaternions.

##  1 Introduction

Quaternion algebras find a wide application in graphics, robotics and control
of spatial rotation of solid bodies, including aerospace flight dynamics [1,
2, 3, 4]. During the last ten years there is a tendency to replace quaternions
by multivectors (MVs) of geometric (aka Clifford) algebras (GAs), mainly due
to the possibility to carry out calculations in higher dimensional GAs of
mixed signatures [5, 6, 7, 8, 9] and, consequently to employ wider GA
capabilities. Of special mention is conformal $\mathit{Cl}_{4,1}$ GA that
allows to do complicated graphics in 5D vector space and then transform the
graphics to 3D Euclidean space for visualization [10].

In this paper we investigate low dimensional algebras from GA perspective,
namely, 1D complex and hyperbolic number algebras as well as 2D algebras
$\mathit{Cl}_{0,2}$, $\mathit{Cl}_{1,1}$ and $\mathit{Cl}_{2,0}$ that are
isomorphic to quaternionic algebras: the Hamilton quaternion (or briefly the
quaternion) [3, 4], coquaternion also known as a split quaternion [11, 12],
and conectorine [13]. The properties of the Hamilton quaternion, which is
isomorphic to $\mathit{Cl}_{0,2}$, recently have been summarized in a handbook
[4]. The coquaternion and conectorine are less known. They are isomorphic to
$\mathit{Cl}_{1,1}$ and $\mathit{Cl}_{2,0}$ algebras that are noncommutative
too. As we shall see, in all 2D algebras the exponential and logarithm may be
treated in a uniform way if they are reformulated in GA terms what, in turn,
helps to generalize and better understand known properties as well as to
discover new ones, for example, continuous degrees of freedom related to a
free vector pointing in an arbitrary direction [14]. In this paper, general
formulas for exponential and logarithm are presented in a form of sum of GA
basis elements and in a basis-free form. The subject considered in this paper
is akin to exponential factorization of MV into product of exponentials [15]
and square root of MV [16, 17].

In Sec. 2, notation and general properties of GA exponential and logarithm
functions are introduced. The GA expressions in 1D are presented in Sec. 3. In
Secs. 4 and 5, respectively, the exponential and logarithm in 2D are
considered. In Addendum (Sec. 6) the square root of MV is discussed. Finally,
in Sec. 7 the conclusion and short discussion are given.

##  2 Properties of exponential and logarithm in GA

Let $\mathbf{e}_{i}$ be the basis vector and
$\mathbf{e}_{ij}\equiv\mathbf{e}_{i}\mathbf{e}_{j}=-\mathbf{e}_{ji}$ be the
bivector. The latter is the geometric product of two orthogonal basis vectors.
Complex and hyperbolic numbers (aka Clifford numbers) in GA [18] are
represented by the following MVs

$\begin{split}\mathit{Cl}_{0,1},\quad&\mathsf{A}=a_{0}+a_{1}I,\quad\text{where\
}I\equiv\mathbf{e}_{1}\ \text{and\ }\mathbf{e}_{1}^{2}=-1,\\\
\mathit{Cl}_{1,0},\quad&\mathsf{A}=a_{0}+a_{1}I,\quad\text{where\
}I\equiv\mathbf{e}_{1}\ \text{and\ }\mathbf{e}_{1}^{2}=+1,\\\ \end{split}$
(2.1)

where $a_{0}$ and $a_{1}$ are the real coefficients,
$a_{0},a_{1}\in\mathbb{R}$. $a_{0}$ is called the scalar part of MV and
$a_{1}I$ the pseudoscalar. In 1D GAs the basis vectors coincide with an
elementary pseudoscalar $I$. The squares,
$\mathbf{e}_{1}^{2}\equiv\mathrm{i}^{2}=-1$ in $\mathit{Cl}_{0,1}$ and
$\mathbf{e}_{1}^{2}=1$ in $\mathit{Cl}_{1,0}$, suggest that we have to do with
complex and hyperbolic numbers, respectively.

In 2-dimensional (2D) algebras there are two basis vectors $\mathbf{e}_{1}$
and $\mathbf{e}_{2}$, and a bivector $\mathbf{e}_{12}\equiv I$ (oriented
plane). The general Clifford number $\mathsf{A}$ is

$\begin{split}&\mathit{Cl}_{0,2}:\
\mathsf{A}=a_{0}+a_{1}\mathbf{e}_{1}+a_{2}\mathbf{e}_{2}+a_{12}I,\
\text{where\ }\ \mathbf{e}_{1}^{2}=\mathbf{e}_{2}^{2}=-1,\ I^{2}=-1,\\\
&\mathit{Cl}_{1,1}:\
\mathsf{A}=a_{0}+a_{1}\mathbf{e}_{1}+a_{2}\mathbf{e}_{2}+a_{12}I,\
\text{where\ }\ \mathbf{e}_{1}^{2}=-\mathbf{e}_{2}^{2}=-1,\ I^{2}=1,\\\
&\mathit{Cl}_{2,0}:\
\mathsf{A}=a_{0}+a_{1}\mathbf{e}_{1}+a_{2}\mathbf{e}_{2}+a_{12}I,\
\text{where\ }\ \mathbf{e}_{1}^{2}=\mathbf{e}_{2}^{2}=+1,\
I^{2}=-1.\end{split}$ (2.2)

The sum $\mathbf{a}=a_{1}\mathbf{e}_{1}+a_{2}\mathbf{e}_{2}$ represents a
general vector in 2D bivector plane. The basis vectors satisfy
$\mathbf{e}_{1}\mathbin{\cdot}\mathbf{e}_{2}=0$ (orthogonality) and
$\mathbf{e}_{1}\mathbin{\wedge}\mathbf{e}_{2}=\mathbf{e}_{1}\mathbf{e}_{2}\equiv\mathbf{e}_{12}$
(oriented unit plane), where the dot and wedge denote the inner and outer
products. $\mathbf{e}_{12}$ plays the role of an elementary pseudoscalar $I$.
The sign of $I^{2}$ depends on algebra, Eq. (2.2). The algebras
$\mathit{Cl}_{1,1}$ and $\mathit{Cl}_{2,0}$ are isomorphic under the following
exchange of GA basis elements: $\mathbf{e}_{1}\leftrightarrow\mathbf{e}_{1}$,
$\mathbf{e}_{2}\leftrightarrow\mathbf{e}_{12}$ and
$\mathbf{e}_{12}\leftrightarrow\mathbf{e}_{2}$.

The main involutions, namely the reversion, inversion and Clifford conjugation
denoted, respectively, by tilde, circumflex and their combination are defined
by the following changes in component signs of MV
$\mathsf{A}=a_{0}+\mathbf{a}+a_{12}I$,

$\begin{split}&\widetilde{\mathsf{A}}=a_{0}+\mathbf{a}-a_{12}I,\qquad\wideparen{\mathsf{A}}=a_{0}-\mathbf{a}+a_{12}I,\qquad\widetilde{\wideparen{\mathsf{A}}}=a_{0}-\mathbf{a}-a_{12}I.\end{split}$
(2.3)

For complex and hyperbolic numbers there is only a single involution,
$\wideparen{\mathsf{A}}=a_{0}+a_{1}\wideparen{I}=a_{0}-a_{1}I$, that usually
is denoted by asterisk in physics and engineering and overline in mathematics.

The exponential of MV is another MV that belongs to the same geometric algebra
$\mathit{Cl}_{p,q}$. If $\mathsf{A}$ and $\mathsf{B}$ are MVs, the following
properties hold:

$\begin{split}&\text{exp}(\mathsf{A}+\mathsf{B})=\text{exp}(\mathsf{A})\text{exp}(\mathsf{B})\quad\text{iff
\ }\mathsf{A}\mathsf{B}=\mathsf{B}\mathsf{A},\\\
&\widetilde{\mathrm{e}^{\mathsf{A}}}=\mathrm{e}^{\widetilde{\mathsf{A}}},\quad\wideparen{\mathrm{e}^{\mathsf{A}}}=\mathrm{e}^{\wideparen{\mathsf{A}}},\quad\widetilde{\wideparen{\mathrm{e}^{\mathsf{A}}}}=\mathrm{e}^{\widetilde{\wideparen{\mathsf{A}}}},\\\
\end{split}$ (2.4)

where $\mathrm{e}$ is the base of the natural logarithm. In 1D algebras the
first property is always satisfied since the commutation of scalar and vector
is satisfied.

The GA exponential can be represented as a power series in a form similar to
scalar exponential [19]. In numerical form, i.e., when coefficients at basis
elements $\mathbf{e}_{1}$, $\mathbf{e}_{1}$ and $\mathbf{e}_{12}$ are real
numbers, the exponential can be summed up approximately [14]. To minimize the
number of multiplications it is convenient to rewrite the exponential in a
nested form (aka Horner’s rule),

$\mathrm{e}^{\mathsf{A}}=1+\frac{\mathsf{A}}{1}(1+\frac{\mathsf{A}}{2}(1+\frac{\mathsf{A}}{3}(1+\frac{\mathsf{A}}{4}(1+\dots)))),$
(2.5)

which requires fewer MV products. If numerical coefficients in $\mathsf{A}$
are small enough the exponential $\mathrm{e}^{\mathsf{A}}$ can be approximated
by truncated series (2.5) to high precision. For examples, refer to paper
[14].

The following properties hold for MV logarithm:

$\begin{split}&\text{log}(\mathsf{A}\mathsf{B})=\text{log}(\mathsf{A})+\text{log}(\mathsf{B})\quad\text{iff\
}\mathsf{A}\mathsf{B}=\mathsf{B}\mathsf{A},\\\
&\mathrm{e}^{-\text{log}(\mathsf{A})}=\mathsf{A}^{-1},\\\
&\widetilde{\log(\mathsf{A})}=\log(\widetilde{\mathsf{A}}),\quad\wideparen{\log(\mathsf{A})}=\log(\wideparen{\mathsf{A}}),\quad\widetilde{\wideparen{\log(\mathsf{A})}}=\mathrm{e}^{\widetilde{\wideparen{\mathsf{A}}}}.\\\
\end{split}$ (2.6)

When the logarithm of MV exists, it may be approximated by series

$\begin{split}\log\mathsf{B}=\mathsf{B}(1+\mathsf{B}(-\tfrac{1}{2}+\mathsf{B}(\tfrac{1}{3}+\mathsf{B}(-\tfrac{1}{4}+\mathsf{B}(\tfrac{1}{5}+\dotsm))))),\quad
0<\lvert\mathsf{B}\rvert<1.\end{split}$ (2.7)

Here $|\mathsf{B}|$ is the determinant norm [14, 20]. If logarithm exists, a
finite series (2.7) can be summed up in a numerical form approximately.
However, as we shall see later there may be sector(s) in the MV coefficient
domain where the logarithm does not exist at all.

In $\mathit{Cl}_{0,1}$ algebra the norm $\lvert\mathsf{B}\rvert$, which is
equal to the square root of MV determinant
$\text{Det}(\mathsf{B})=\mathsf{B}\wideparen{\mathsf{B}}=b_{0}^{2}+b_{1}^{2}>0$,
is called the magnitude or absolute value of the MV (or magnitude of the
complex number in this case). In hyperbolic number theory similar role is
played by product $\mathsf{B}\wideparen{\mathsf{B}}=b_{0}^{2}-b_{1}^{2}$,
which may be positive, negative or zero. In this case the magnitude called a
determinant semi-norm (or pseudonorm)
$\lVert\mathsf{B}\rVert=\sqrt{\operatorname{abs}(\mathsf{B}\wideparen{\mathsf{B}})}=\sqrt{\operatorname{abs}(b_{0}^{2}-b_{1}^{2})}\geq
0$ is introduced. Note that now the equality sign appears, therefore, the
semi-norm $\lVert\mathsf{B}\rVert$ may be zero even if $\mathsf{B}\neq 0$. The
equality sign in case of the norm $\lvert\mathsf{B}\rvert$ would require the
MV to nullify.

a) b)

Figure 1:  Analogy between unit circle $x^{2}+y^{2}=1$ and unit hyperbola
$x^{2}-y^{2}=1$. For a circle the coordinates of point $b$ are
$x=\cos\varphi_{c}$ and $y=\sin\varphi_{c}$, and for a hyperbola they are
$x=\cosh\varphi_{h}$ and $y=\sinh\varphi_{h}$. The trigonometric and
hyperbolic angles are defined, respectively, by $\varphi_{c}=\arctan(y/x)$ in
the range $\varphi_{c}=[0,2\pi)$ and $\varphi_{h}=\textrm{artanh}(y/x)$ in the
range $\varphi_{h}=(-\infty,+\infty)$. For hyperbola they are limited by
asymptotes (dashed lines). The shaded areas $A_{c}$ and $A_{h}$ are
proportional to trigonometric and hyperbolic angles: $A_{c}=\varphi_{c}/2$ and
$A_{h}=\varphi_{h}/2$. The infinity signs at asymptotes show extreme values of
$\varphi_{h}/2$, where the infinities having opposite signs meet,
$-\infty=+\infty$ [21].

##  3 Exponential and logarithm in 1D algebras

One-dimensional GAs are represented by two commutative algebras: the well-
known complex number algebra which is isomorphic to $\mathit{Cl}_{0,1}$ and
the hyperbolic number algebra $\mathit{Cl}_{1,0}$ [18]. In Fig. 1 the
geometrical properties of both algebras are compared graphically on $x$-$y$
plane (equivalently on $b_{0}$-$b_{1}$ plane). In Fig. 1b the two branches of
hyperbola close down at plus/minus infinities [21]. The shaded area in both
cases is proportional either to inner $\varphi_{c}$ or outer $\varphi_{h}$
angle between the center and the point $b$ on circle $y^{2}+x^{2}=1$ or
hyperbola $y^{2}-x^{2}=1$, respectively. If a point $b$ on the circle or
hyperbola represents the MV $\mathsf{B}=b_{0}+b_{1}\mathbf{e}_{1}$ then in GA
the quantity
$\mathsf{B}\wideparen{\mathsf{B}}=b_{0}^{2}+b_{1}^{2}=\lvert\mathsf{B}\rvert^{2}>0$
is the square of norm that graphically represents the sector $A_{c}$ in Fig.
1a. Similarly the sector $A_{h}$ in Fig. 1b represents the semi-norm
$\lVert\mathsf{B}\rVert$ (pseudonorm) that as mentioned may be positive,
negative, or zero.

###  3.1 Exponential and logarithm of MV in $\mathit{Cl}_{0,1}$

Since $\mathit{Cl}_{0,1}$ algebra is commutative and $\mathbf{e}_{1}^{2}=-1$,
we can write

$\mathrm{e}^{\mathsf{B}}=\mathrm{e}^{b_{0}+b_{1}\mathbf{e}_{1}}=\mathrm{e}^{b_{0}}\mathrm{e}^{b_{1}\mathbf{e}_{1}}=\mathrm{e}^{b_{0}}\big{(}\cos
b_{1}+\mathbf{e}_{1}\sin b_{1}\big{)},$ (3.1)

where Euler’s rule was used. Presence of trigonometric functions indicates
that the exponential in $\mathit{Cl}_{0,1}$ is a periodic function with period
$2\pi k$, where $k\in\mathbb{Z}$ is an arbitrary integer. Thus, more generally
in $\mathit{Cl}_{0,1}$ we have
$\mathrm{e}^{\mathsf{B}}=\mathrm{e}^{b_{0}+b_{1}\mathbf{e}_{1}+2\pi
k\mathbf{e}_{1}}$.

The logarithm of a complex number $z=x+\mathrm{i}y$ is

$\log z=\log(r\mathrm{e}^{\mathrm{i}\varphi})=\log\lvert
z\rvert+\mathrm{i}\varphi=\log\big{(}\sqrt{x^{2}+y^{2}}\big{)}+\mathrm{i}\varphi,$
(3.2)

which in $\mathit{Cl}_{0,1}$ algebra notation is

$\log\mathsf{B}=\log\sqrt{b_{0}^{2}+b_{1}^{2}}+\mathbf{e}_{1}\arctan(b_{1}/b_{0})=\log\lvert\mathsf{B}\rvert+\mathbf{e}_{1}\varphi.$
(3.3)

The angle $\varphi=\arctan(y/x)$, or $\varphi=\arctan(b_{1}/b_{0})$, is called
the argument of logarithm. If $r=(b_{0}^{2}+b_{1}^{2})^{1/2}$ is const. then
$\varphi$ may be interpreted as a rotation angle of a vector around coordinate
center, Fig. 2a. To eliminate sign ambiguity between quadrants 1 and 3 (or 2
and 4), the arc tangent of a single argument usually is replaced by double
argument arc tangent $\arctan(x,y)$. If signs of $x$ and $y$ are already fixed
then $\arctan(x,y)=\arctan(y/x)$. To include multiple rotations, after every
single rotation the period $2\pi$ is added to $\varphi$, so that after $k$
rotations we have $k$-windings in Fig. 2b and angle
$\varphi=\text{arctan}(b_{0},b_{1})+2\pi k$, where
$k\in\mathbb{Z}=\dots-2,-1,0,1,2,\dots$. Similarly, in case of hyperbolic
functions to include the sign of $x$ and $y$ in the quadrants 1-4 one may
introduce a double angle hyperbolic tangent***Figure 1b represents properties
of hyperbola drawn on the Euclidean plane. The properties of hyperbola on
sphere and complex cylinder are described in [21, 22]. Note that Mathematica
computes the area hyperbolic tangent of real or complex argument on a complex
closed cylinder [21].
$\text{tanh}(x,y)=y/x=\sinh\varphi_{h}/\cosh\varphi_{h}=\tanh\varphi_{h}$. As
follows from Fig. 1b the range of the hyperbolic tangent is $(-1...1)$ when
${\varphi_{h}=-\infty...\infty}$. Then, in the quadrants 1,2,3,4 we have,
respectively, $\text{tanh}(+|x|,+|y|)$, $\text{tanh}(-|x|,+|y|)$,
$\text{tanh}(-|x|,-|y|)$, and $\text{tanh}(+|x|,-|y|)$.

Figure 2:  Graphical representation of $\mathit{Cl}_{0,1}$ logarithm. The real
part $\text{Re}(\log(\mathsf{B}))=\log(\sqrt{b_{0}^{2}+b_{1}^{2}})$ (left
panel) and principal logarithm $\varphi=\text{Im}(\log(\mathsf{B}))$ in the
range $[-\pi,\pi]$ is represented by a single winding on the right panel. At a
fixed $\varphi$ the lines run parallel to horizontal $\\{b_{0},b_{1}\\}$
plane.

We shall assume that in GA the defining equation of the logarithm is
$\log\mathsf{B}=\mathsf{A}$, which takes into account only the principal value
(principal logarithm). To include multiple values we add a free MV
$\mathsf{F}$,

$\log\mathsf{B}=\mathsf{A}+\mathsf{F},\qquad\mathsf{A},\mathsf{B},\mathsf{F}\in\mathit{Cl}_{0,1},$
(3.4)

that satisfies $\mathrm{e}^{\mathsf{F}}=1$. Equation (3.4) is more general
because, as we shall see, it allows to include the multiplicity into GA
logarithm in case of higher ($n=3$) dimensional GAs [20]. Let’s apply the
described approach to Eq. (3.3)

$\log\mathsf{B}=\text{log}\lvert\mathsf{B}\rvert+\mathbf{e}_{1}\text{arctan}(b_{0},b_{1})\equiv\log
r+\mathbf{e}_{1}\varphi,$ (3.5)

where
$\lvert\mathsf{B}\rvert=\sqrt{\mathsf{B}\wideparen{\mathsf{B}}}=\sqrt{b_{0}^{2}+b_{1}^{2}}=r$
is the radius $r$ (magnitude or norm of $\mathsf{B}$) and $\varphi$ is the
angle between the horizontal axis and line that connects the coordinate center
with the point $b$, Fig. 1a. To include multiplicity in the angle, a free term
$\mathsf{F}$ is added, $\log\mathsf{B}=\mathsf{A}+\mathsf{F}$. After
substitution of $\mathsf{F}=f_{0}+\mathbf{e}_{1}f_{1}$ into
$\mathrm{e}^{\mathsf{F}}=1$ and using the trigonometric expansion similar to
Eq. (3.1), we find $\mathrm{e}^{f_{0}}(\cos f_{1}+\mathbf{e}_{1}\sin
f_{1})=1$, the solution of which is $f_{0}=0$ and $f_{1}=2\pi k$, where
$k\in\mathbb{Z}$. Thus, the full solution in agreement with the complex
function theory can be written

$\mathit{Cl}_{0,1}:\quad\log\mathsf{B}=\text{log}\lvert\mathsf{B}\rvert+\mathbf{e}_{1}(\varphi+2\pi
k),\quad 0\leq\varphi<2\pi,\quad k\in\mathbb{Z}.$ (3.6)

At a fixed $r=\lvert\mathsf{B}\rvert$ this equation represents the spiral with
period $2\pi$ since the argument ($0\leq\varphi<2\pi$) increases by $2\pi$
after every single winding in the “complex” plane $\\{b_{0},b_{1}\\}$. The
logarithm (3.3) exists for all values of $\mathsf{B}$. Often it is assumes
that the principal logarithm is in the range $-\pi<\varphi<\pi$, then the
logarithm is

$\mathit{Cl}_{0,1}:\quad\log\mathsf{B}=\begin{cases}\text{log}\lvert\mathsf{B}\rvert+\mathbf{e}_{1}\varphi&\quad\text{if\
}b_{0}>0\text{\ and\ }b_{1}\neq 0,\\\
\text{log}\lvert\mathsf{B}\rvert+\mathbf{e}_{1}(\varphi+\pi)&\quad\text{if\
}b_{0}<0\text{\ and\ }b_{1}>0,\\\
\text{log}\lvert\mathsf{B}\rvert+\mathbf{e}_{1}(\varphi-\pi)&\quad\text{if\
}b_{0}<0\text{\ and\ }b_{1}<0,\\\ \hline\cr b_{0}&\quad\text{if\
}b_{0}>0\text{\ and\ }b_{1}=0,\\\ b_{0}+\mathbf{e}_{1}\pi&\quad\text{if\
}b_{0}<0\text{\ and\ }b_{1}=0,\\\ b_{1}+\mathbf{e}_{1}\pi/2&\quad\text{if\
}b_{0}=0\text{\ and\ }b_{1}>0,\\\ b_{1}-\mathbf{e}_{1}\pi/2&\quad\text{if\
}b_{0}=0\text{\ and\ }b_{1}<0.\\\ \end{cases}$ (3.7)

The first three expressions are the main formulas. The remaining represent
special cases: they show the behavior of logarithm on the real and imaginary
axis. When $b_{0}=b_{1}=0$ the logarithm is undefined. The definition given by
Eqs (3.7) and visualized in Fig. 2b frequently is met in applications. It has
been accepted in ISO standards such as C programming language and Mathematica.

###  3.2 Exponential and logarithm of MV in $\mathit{Cl}_{1,0}$

For 1D algebras the inverse of MV $\mathsf{B}=b_{0}+b_{1}\mathbf{e}_{1}$ is

$\mathit{Cl}_{0,1}:\
\mathsf{B}^{-1}=\frac{\wideparen{\mathsf{B}}}{\mathsf{B}\wideparen{\mathsf{B}}}=\frac{b_{0}-\mathbf{e}_{1}b_{1}}{b_{0}^{2}+b_{1}^{2}};\qquad\mathit{Cl}_{1,0}:\
\mathsf{B}^{-1}=\frac{\wideparen{\mathsf{B}}}{\mathsf{B}\wideparen{\mathsf{B}}}=\frac{b_{0}-\mathbf{e}_{1}b_{1}}{b_{0}^{2}-b_{1}^{2}},$
(3.8)

and satisfies $\mathsf{B}^{-1}\mathsf{B}=\mathsf{B}\mathsf{B}^{-1}=1$. From
(3.8) follows that, in contrast to complex algebra where each nonzero complex
number has its inverse, in $\mathit{Cl}_{1,0}$ nonzero divisors appear if
$b_{0}^{2}=b_{1}^{2}$ as shown by dashed lines in a hyperbolic plane in Fig.
3. Since $\mathbf{e}_{1}^{2}>0$, $\mathit{Cl}_{1,0}$ exponential may be
expanded in hyperbolic sine and cosine functions [19],

$\mathit{Cl}_{1,0}:\quad\mathrm{e}^{\mathsf{B}}=\mathrm{e}^{b_{0}+b_{1}\mathbf{e}_{1}}=\mathrm{e}^{b_{0}}\mathrm{e}^{b_{1}\mathbf{e}_{1}}=\mathrm{e}^{b_{0}}\big{(}\cosh
b_{1}+\mathbf{e}_{1}\sinh b_{1}\big{)},$ (3.9)

where $b_{0},b_{1}\in\mathbb{R}$. The hyperbolic functions are monotonic
therefore the exponential in $\mathit{Cl}_{1,0}$ inherits this property as
well.

Figure 3: Hyperbolic plane $\\{b_{0},b_{1}\\}$ that represents
$\mathit{Cl}_{1,0}$ algebra. The arrows show MV
$\mathsf{B}=b_{0}+\mathbf{e}_{1}b_{1}$ and its conjugate
$\wideparen{\mathsf{B}}$. The dashed lines are asymptotes
$b_{0}^{2}-b_{1}^{2}=0$. The principal square root and logarithm exist in the
shaded sector where $b_{0}>b_{1}$. I-IV are the hyperbolic plane quadrants.

In $\mathit{Cl}_{1,0}$ a logarithm defining equation is
$\log\mathsf{B}=\mathsf{A}$, in solution of which the hyperbolic functions and
the identity $\cosh^{2}x-\sinh^{2}x=1$ are to be used. The following
expression for principal logarithm (the first formula) and special case (the
second formula ) is found:

$\mathit{Cl}_{1,0}:\quad\log\mathsf{B}=\begin{cases}\log\sqrt{b_{0}^{2}-b_{1}^{2}}+\mathbf{e}_{1}\text{artanh}(b_{1}/b_{0});\
b_{0}>0\ \text{and}\ b_{0}^{2}>b_{1}^{2},\\\
\frac{1}{2}(\log(0_{+}{})+\log(2b_{0}))\pm\mathbf{e}_{1}\frac{1}{2}(-\log(0_{+})+\log(2b_{0});\
b_{1}=b_{0},\end{cases}$ (3.10)

where artanh is the area tangent function, $-1<\text{artanh}(b_{1}/b_{0})<1$.
The scalar part $\log\sqrt{b_{0}^{2}-b_{1}^{2}}$ exists if
$b_{0}^{2}>b_{1}^{2}$. The logarithm has a genuine value if a pair
$\\{b_{0},b_{1}\\}$ is in the shaded sector of Fig. 3. Thus, the existence of
both the logarithm and the square root are determined by condition
$b_{0}^{2}>b_{2}^{2}$. The special case belongs to asymptotes $b_{0}=b_{1}$,
where $\log(0_{+})$ is the logarithm of a point infinitesimally close to zero.
This term vanishes in $\exp{(\log{\mathsf{B}})}=\mathsf{B}$ (see Example 2).
The first equation of (3.10) can be rewritten in hyperbola parameters in Fig.
1b. Since $\mathsf{B}=r(\cosh\varphi_{h}+\mathbf{e}_{1}\sinh\varphi_{h})$,
where $r=a_{0}$ is the radius ($r=1$ in Fig. 1b) we have
$b_{0}=r\cosh\varphi_{h}$ and $b_{1}=r\sinh\varphi_{h}$. Since
$r=\sqrt{b_{0}^{2}-b_{1}^{2}}$ and $\tanh\varphi_{h}=b_{1}/b_{0}$, we have
that $\log\mathsf{B}=\log r+\mathbf{e}_{1}\varphi_{h}$ which is to be compared
with Eq. (3.3).

To find the free term $\mathsf{F}$ we solve $\exp({\mathsf{F}})=1$, for this
purpose bringing into play Eq. (3.9),

$\mathrm{e}^{\mathsf{F}}=\mathrm{e}^{f_{0}+\mathbf{e}_{1}f_{1}}=\mathrm{e}^{f_{0}}\big{(}\cosh\lvert
f_{1}\rvert+\mathbf{e}_{1}\sinh{\lvert f_{1}\rvert}\big{)}=1.$ (3.11)

This equation can be satisfied if $f_{0}=f_{1}=0$. So, in this algebra we have
only the principal logarithm.

Example 1. $\mathit{Cl}_{1,0}$: If $b_{0}^{2}>b_{1}^{2}$, $b_{0}>0$, and
$\mathsf{B}=3\pm 2\mathbf{e}_{1}$ then
$\log\mathsf{B}=\log\sqrt{5}\pm\mathbf{e}_{1}\text{artanh}\frac{2}{3}=a_{0}\pm\mathbf{e}_{1}a_{1}$.
The exponential of logarithm gives $\exp(\log\mathsf{B})=\mathsf{B}$. If
$b_{0}^{2}>b_{1}^{2}$, $b_{0}<0$ and $\mathsf{B}^{\prime}=-3\pm
2\mathbf{e}_{1}$, then
$\log\mathsf{B}^{\prime}=\log\sqrt{5}\pm\mathbf{e}_{1}\text{artanh}\frac{2}{3}=a_{0}\pm\mathbf{e}_{1}a_{1}\,$.
The answer is wrong since the initial MV is returned with an opposite sign:
$\exp(\log\mathsf{B}^{\prime})=-\mathsf{B}^{\prime}$.

Example 2. $\mathit{Cl}_{1,0}$: $b_{0}=b_{1}$, $\mathsf{B}=2+2\mathbf{e}_{1}$.
In this case the second formula of (3.10) should be used:
$\log\mathsf{B}=\frac{1}{2}(\log 4+\log 0_{+})+\frac{1}{2}\mathbf{e}_{1}(\log
4-\log 0_{+})$. Then
$\exp(\log\mathsf{B})=\frac{1}{2}\big{(}4(1+\mathbf{e}_{1})\big{)}+\frac{1}{2}\big{(}\mathrm{e}^{\log
0_{+}}(-1+\mathbf{e}_{1})\big{)}\to 2(1+\mathbf{e}_{1})$ which in the limit
$\log 0_{+}\to-\infty$ gives $\mathsf{B}=2+2\mathbf{e}_{1}$ that represents a
point on the asymptote.

##  4  Exponential and logarithm in 2D algebras

###  4.1 Quaternionic ‘vector’

The following defining equations for exponential and logarithm in 2D GAs are
used, $\exp\mathsf{B}=\mathsf{A}$ and $\log\mathsf{B}=\mathsf{A}$, where MVs
$\mathsf{A}$ and $\mathsf{B}$ belong to the same algebra. It is convenient to
introduce base-free MVs $A^{\prime}=\mathbf{a}+b_{12}\mathbf{e}_{12}$ and
$B^{\prime}=\mathbf{b}+b_{12}\mathbf{e}_{12}$, where $\mathbf{a}$ and
$\mathbf{b}$ are vectors in $\mathbf{e}_{12}$ plane. In analogy to Hamilton
quaternion theory [3, 4], in the following we shall treat the quantity
$A^{\prime}$ as a 3D ‘vector’. Introduction of such a ‘vector’ appears very
helpful in calculating the exponential as well as logarithm in all 2D
algebras. Thus, a full MV in 2D algebras may be represented as a sum of scalar
and ‘vector’:

$\mathsf{A}=a_{0}+A^{\prime},\quad
A^{\prime}=a_{1}\mathbf{e}_{1}+a_{2}\mathbf{e}_{2}+a_{12}\mathbf{e}_{12}.$
(4.1)

In $\mathit{Cl}_{0,2}$ the squares of all three basis elements satisfy
$\mathbf{e}_{1}^{2}=\mathbf{e}_{2}^{2}=\mathbf{e}_{12}^{2}=-1$ and
$\mathbf{e}_{1}\mathbf{e}_{2}\mathbf{e}_{12}=-1$. Similarly, in the Hamilton
quaternion algebra [3, 4] a set of three imaginary units
$\\{\mathrm{i},\mathrm{j},\mathrm{k}\\}$ satisfy
$\mathrm{i}^{2}=\mathrm{j}^{2}=\mathrm{k}^{2}=-1$ and
$\mathrm{i}\mathrm{j}\mathrm{k}=-1$. In 3D Euclidean space the quaternionic
vector is defined by
$\mathbf{v}=a_{1}\mathrm{i}+a_{2}\mathrm{j}+a_{12}\mathrm{k}$, the square of
which is a negative number. The same property is satisfied by ‘vector’,
$(A^{\prime})^{2}\equiv A^{\prime 2}=-a_{1}^{2}-a_{2}^{2}-a_{12}^{2}<0$, where
$a_{1},\ a_{2}$ and $a_{12}$ are the real numbers. Thus, the MV $A^{\prime}$
is equivalent to quaternion vector and $A^{\prime}$ may be treated exactly in
the same way as the Hamilton vector
$\mathbf{v}=a_{1}\mathrm{i}+a_{2}\mathrm{j}+a_{12}\mathrm{k}$. For
$\mathit{Cl}_{0,2}$ ’vector’ $A^{\prime}$ the norm is defined by $\lvert
A^{\prime}\rvert=\sqrt{-A^{\prime 2}}$.

Because $\mathit{Cl}_{1,1}$ and $\mathit{Cl}_{2,0}$ are not division algebras,
i.e. in these algebras not every MV has inverse, for these algebras we have
different cases. Now $A^{\prime 2}\equiv(A^{\prime})^{2}$ may be either
positive or negative, or even zero. The first (positive) case, as we shall
see, is related to hyperbolic functions, while the second is related to
trigonometric functions. Both cases will be investigated separately in
$\mathit{Cl}_{1,1}$ and $\mathit{Cl}_{2,0}$ algebras. The semi-norm of
’vector’ $A^{\prime}$ is defined by $\lVert
A^{\prime}\rVert=\sqrt{\operatorname{abs}(A^{\prime 2})}\geq 0$.

|  $\exp(\mathsf{B})=\exp(b_{0}+B^{\prime})$ $-B^{\prime}=b_{1}\mathbf{e}_{1}+b_{2}\mathbf{e}_{2}+b_{12}\mathbf{e}_{12}$ |
---|---|---
$\mathit{Cl}_{0,2}$ | $\begin{cases}\mathrm{e}^{b_{0}}\Big{(}\cos\lvert B^{\prime}\rvert+\frac{B^{\prime}}{\lvert B^{\prime}\rvert}\sin\lvert B^{\prime}\rvert\Big{)}&\qquad\quad\ B^{\prime 2}=b_{1}^{2}+b_{2}^{2}+b_{12}^{2}\,>0\\\ \mathrm{e}^{b_{0}}&\qquad\quad\ B^{\prime}=0\end{cases}$ |
$\mathit{Cl}_{1,1}$ | $\begin{cases}\mathrm{e}^{b_{0}}\Big{(}\cosh\lVert B^{\prime}\rVert+\frac{B^{\prime}}{\lVert B^{\prime}\rVert}\sinh\lVert B^{\prime}\rVert\Big{)}&\quad B^{\prime 2}=b_{1}^{2}-b_{2}^{2}+b_{12}^{2}\,>0\\\ \mathrm{e}^{b_{0}}(1+B^{\prime})&\quad B^{\prime 2}=b_{1}^{2}-b_{2}^{2}+b_{12}^{2}=0\\\ \mathrm{e}^{b_{0}}\Big{(}\cos\lVert B^{\prime}\rVert+\frac{B^{\prime}}{\lVert B^{\prime}\rVert}\sin\lVert B^{\prime}\rVert\Big{)}&\quad B^{\prime 2}=b_{1}^{2}-b_{2}^{2}+b_{12}^{2}\,<0\end{cases}$ |
$\mathit{Cl}_{2,0}$ | $\begin{cases}\mathrm{e}^{b_{0}}\Big{(}\cosh\lVert B^{\prime}\rVert+\frac{B^{\prime}}{\lVert B^{\prime}\rVert}\sinh\lVert B^{\prime}\rVert\Big{)}&\quad B^{\prime 2}=b_{1}^{2}+b_{2}^{2}-b_{12}^{2}\,>0\\\ \mathrm{e}^{b_{0}}(1+B^{\prime})&\quad B^{\prime 2}=b_{1}^{2}+b_{2}^{2}-b_{12}^{2}=0\\\ \mathrm{e}^{b_{0}}\Big{(}\cos\lVert B^{\prime}\rVert+\frac{B^{\prime}}{\lVert B^{\prime}\rVert}\sin\lVert B^{\prime}\rVert\Big{)}&\quad B^{\prime 2}=b_{1}^{2}+b_{2}^{2}-b_{12}^{2}\,<0\end{cases}$ |
Table 1:  Exponentials of general MV
$\mathsf{B}=b_{0}+B^{\prime}=b_{0}+b_{1}\mathbf{e}_{1}+b_{2}\mathbf{e}_{2}+b_{12}\mathbf{e}_{12}$
in 2D GAs. $\lvert B^{\prime}\rvert=\sqrt{-B^{\prime 2}}$ and $\lVert
B^{\prime}\rVert=\sqrt{\operatorname{abs}(B^{\prime 2})}$ are real numbers
that represent the norm and semi-norm, respectively. The trigonometric
functions appear when $B^{\prime 2}<0$ while hyperbolic when $B^{\prime 2}>0$.
Zero values of the semi-norm corresponds to $\lim_{x\to
0}\tfrac{\sin(x)}{x}=\lim_{x\to 0}\tfrac{\sinh(x)}{x}=1$.

###  4.2  Exponentials of MV in 2D algebras

In Table 1, two-dimensional exponentials in expanded form including the case
of null MV (when $\mathsf{B}^{2}=0$) are summarized. The structure of the
formulas reminds de Moivre’s-Euler’s rules. For $\mathit{Cl}_{0,2}$ only
trigonometric functions appear. For algebras $\mathit{Cl}_{1,1}$ and
$\mathit{Cl}_{2,0}$ also hyperbolic functions appear if $B^{\prime 2}>0$. In
case of $\mathit{Cl}_{0,2}$ which is isomorphic to Hamilton quaternion the
exponential formula can be found easily if the property
$\bigl{(}B^{\prime}/\lvert B^{\prime}\rvert\bigr{)}^{2}=-1$ is taken into
account. Since $B^{\prime}/\lvert B^{\prime}\rvert$ behaves like an imaginary
unit we can write at once

$\mathrm{e}^{B^{\prime}}=\cos\lvert B^{\prime}\rvert+\frac{B^{\prime}}{\lvert
B^{\prime}\rvert}\sin\lvert B^{\prime}\rvert.$ (4.2)

Then, the exponential of $\mathsf{B}=b_{0}+B^{\prime}$ is

$\mathit{Cl}_{0,2}:\
\mathrm{e}^{\mathsf{B}}=\mathrm{e}^{b_{0}+B^{\prime}}=\mathrm{e}^{b_{0}}\mathrm{e}^{B^{\prime}}=\mathrm{e}^{b_{0}}\bigl{(}\cos\lvert
B^{\prime}\rvert+\frac{B^{\prime}}{\lvert B^{\prime}\rvert}\sin\lvert
B^{\prime}\rvert\bigr{)}.$ (4.3)

In the remaining algebras the different normalization must be used. The square
of a normalized ‘vector’ now is $\bigl{(}B^{\prime}/\lVert
B^{\prime}\rVert\bigr{)}^{2}=\pm 1$ and apart from trigonometric, in addition,
hyperbolic functions for plus sign appear,

$\mathit{Cl}_{1,1},\mathit{Cl}_{2,0}:\
\mathrm{e}^{\mathsf{B}}=\mathrm{e}^{b_{0}}\mathrm{e}^{B^{\prime}}=\begin{cases}\mathrm{e}^{b_{0}}\bigl{(}\cos\lVert
B^{\prime}\rVert+\frac{B^{\prime}}{\lVert B^{\prime}\rVert}\sin\lVert
B^{\prime}\rVert\bigr{)},\quad B^{\prime 2}<0,\\\
\mathrm{e}^{b_{0}}\bigl{(}\cosh\lVert
B^{\prime}\rVert+\frac{B^{\prime}}{\lVert B^{\prime}\rVert}\sinh\lVert
B^{\prime}\rVert\bigr{)},\quad B^{\prime 2}>0.\end{cases}$ (4.4)

Thus, in $\mathit{Cl}_{1,1}$ and $\mathit{Cl}_{2,0}$ algebras depending on
sign of $B^{\prime 2}$ and coefficient values in the semi-norm, the
exponentials may be expanded either in trigonometric or in hyperbolic
functions and as a result mat be periodic or monotonic. Finally, in Table 1
the exponential
$\mathrm{e}^{\mathsf{B}}=\mathrm{e}^{b_{0}}\big{(}1+B^{\prime}\big{)}$ comes
from the null MV the square of which nullifies $B^{\prime 2}=0$ and yields a
linearly dependence on $B^{\prime}$. Recently we have found [14] that in
three-dimensional GAs (and probably in higher dimensional spaces) the
entanglement or mixing of vector and bivector components may tale place, so
that in the expanded form the exponential loses de Moivre’s-Euler’s formula
structure. The latter is regained if both the vector and bivector lie in the
same plane. This is in agreement with the present 2D formulas where the vector
and bivector are always in $\mathbf{e}_{1}-\mathbf{e}_{2}$ plane.

Example 3.
$\mathsf{B}=2+5\mathbf{e}_{1}-4\mathbf{e}_{2}-7\mathbf{e}_{12}=2+B^{\prime}$.
$\mathit{Cl}_{0,2}$:
$\exp\mathsf{B}=\mathrm{e}^{2}\left(\cos\sqrt{90}+\frac{5\mathbf{e}_{1}-4\mathbf{e}_{2}-7\mathbf{e}_{12}}{\sqrt{90}}\sin\sqrt{90}\right),B^{\prime
2}=-90,\lvert B^{\prime}\rvert=\sqrt{90}$,
$\mathit{Cl}_{1,1}$:
$\exp\mathsf{B}=\mathrm{e}^{2}\left(\cosh\sqrt{58}+\frac{5\mathbf{e}_{1}-4\mathbf{e}_{2}-7\mathbf{e}_{12}}{\sqrt{58}}\sinh\sqrt{58}\right),\
B^{\prime 2}=58,\lVert B^{\prime}\lVert=\sqrt{58}$,
$\mathit{Cl}_{2,0}$:
$\exp\mathsf{B}=\mathrm{e}^{2}\left(\cos\sqrt{8}+\frac{5\mathbf{e}_{1}-4\mathbf{e}_{2}-7\mathbf{e}_{12}}{\sqrt{8}}\sin\sqrt{8}\right),\quad
B^{\prime 2}=-8,\lVert B^{\prime}\rVert=\sqrt{8}$.

###  4.3 Products of exponentials

Using Table 1 it is easy to calculate the geometric product of two
exponentials. For example, for trigonometric functions in $\mathit{Cl}_{0,2}$,
when $B^{\prime 2}<0$, we find

$\begin{split}&\mathrm{e}^{\mathsf{A}}\mathrm{e}^{\mathsf{B}}=\mathrm{e}^{a_{0}+A^{\prime}}\mathrm{e}^{b_{0}+B^{\prime}}=\mathrm{e}^{a_{0}+b_{0}}\left(\cos\lvert
A^{\prime}\rvert\cos\lvert B^{\prime}\rvert+\frac{\langle
A^{\prime}B^{\prime}\rangle_{0}}{\lvert A^{\prime}\rvert\lvert
B^{\prime}\rvert}\sin\lvert A^{\prime}\rvert\sin\lvert
B^{\prime}\rvert\right)+\\\
&\mathrm{e}^{a_{0}+b_{0}}\left(\frac{A^{\prime}}{\lvert
A^{\prime}\rvert}\sin\lvert A^{\prime}\rvert\cos\lvert
B^{\prime}\rvert+\frac{B^{\prime}}{\lvert B^{\prime}\rvert}\sin\lvert
B^{\prime}\rvert\cos\lvert
A^{\prime}\rvert+\frac{1}{2}\frac{[A^{\prime},B^{\prime}]}{\lvert
A^{\prime}\rvert\lvert B^{\prime}\rvert}\sin\lvert A^{\prime}\rvert\sin\lvert
B^{\prime}\rvert\right).\end{split}$ (4.5)

When $A^{\prime}=B^{\prime}$ the commutator $[A^{\prime},B^{\prime}]=0$ and
the Eq (4.5) reduces to double $2\mathsf{A}$ argument exponential. For
remaining algebras the norm should be replaced by semi-norm. Below, particular
cases follow from (4.5).

Case 1. Product of vectorial exponentials. If $\mathsf{A}$ and $\mathsf{B}$
represent vectors $\mathbf{a}=a_{1}\mathbf{e}_{1}+a_{2}\mathbf{e}_{2}$, and
$\mathbf{b}=b_{1}\mathbf{e}_{1}+b_{2}\mathbf{e}_{2}$, then

$\begin{split}\mathrm{e}^{\mathbf{a}}\mathrm{e}^{\mathbf{b}}=&\cos|\mathbf{a}|\cos|\mathbf{b}|-\cos\theta\sin|\mathbf{a}|\sin|\mathbf{b}|+\\\
&\frac{\mathbf{a}}{|\mathbf{a}|}\sin|\mathbf{a}|\cos|\mathbf{b}|+\frac{\mathbf{b}}{|\mathbf{b}|}\sin|\mathbf{b}|\cos|\mathbf{a}|+\mathbf{e}_{12}\sin\theta\sin|\mathbf{a}|\sin|\mathbf{b}|,\end{split}$
(4.6)

where $\theta$ is the angle between vectors $\mathbf{a}$ and $\mathbf{b}$.

Case 2. Product of bivectorial exponentials. If $\mathsf{A}$ and $\mathsf{B}$
are simple bivectors $\mathcal{A}=a_{12}\mathbf{e}_{12}$,
$\mathcal{B}=b_{12}\mathbf{e}_{12}$, then

$\mathrm{e}^{\mathcal{A}}\mathrm{e}^{\mathcal{B}}=\mathrm{e}^{\mathcal{A}+\mathcal{B}}=\cos\lvert\mathcal{A}+\mathcal{B}\rvert+\mathbf{e}_{12}\sin\lvert\mathcal{A}+\mathcal{B}\rvert,$
(4.7)

where
$\lvert\mathcal{A}+\mathcal{B}\rvert=\sqrt{(\mathcal{A}+\mathcal{B})\widetilde{(\mathcal{A}+\mathcal{B})}}$.
Since $\mathcal{A}+\mathcal{B}$ is the bivector and
$(\mathcal{A}+\mathcal{B})^{2}<0$ this formula follows directly.

Case 3. Product of vector and bivector exponentials:

$\mathrm{e}^{\mathbf{a}}\mathrm{e}^{\mathcal{B}}=(\cos{|\mathbf{a}|}+\frac{\mathbf{a}}{|\mathbf{a}|}\sin{|\mathbf{a}}|)(\cos|\mathcal{B}|+\mathbf{e}_{12}\sin|\mathcal{B}|)$
(4.8)

Case 4. The commutator also vanishes if the coefficients satisfy:
$a_{1}b_{12}=b_{1}a_{12}$, $a_{2}b_{12}=b_{2}a_{12}$ and
$a_{2}b_{1}=b_{2}a_{1}$. Since in this case
$[\mathsf{A},\mathsf{B}]=[A^{\prime},B^{\prime}]$ we have
$\mathrm{e}^{\mathsf{A}}\mathrm{e}^{\mathsf{B}}=\mathrm{e}^{\mathsf{A}+\mathsf{B}}$
and Eq. (4.5) reduced to

$\mathrm{e}^{\mathsf{A}}\mathrm{e}^{\mathsf{B}}=\mathrm{e}^{\mathsf{A}+\mathsf{B}}=\mathrm{e}^{\mathsf{B}}\mathrm{e}^{\mathsf{A}}=\mathrm{e}^{a_{0}+b_{0}}\Big{(}\cos\lvert
A^{\prime}+B^{\prime}\rvert+\frac{A^{\prime}+B^{\prime}}{\lvert
A^{\prime}+B^{\prime}\rvert}\sin\lvert A^{\prime}+B^{\prime}\rvert\Big{)}.$
(4.9)

When $B^{{}^{\prime}2}>0$, similar formulas exist for hyperbolic functions.

##  5 Logarithm of MV in 2D algebras

The approach to commutative algebras in Subsec. 3.1 here is generalized to 2D
algebras. The ’vector’ property $B^{\prime 2}\gtreqqless 0$ allows to get 2
dimensional logarithm formulas that are very similar to those found in 1D case
but with basis vector $\mathbf{e}_{1}$ replaced by unit multivector
$B^{\prime}/\lvert B^{\prime}\rvert$ or $B^{\prime}/\lVert B^{\prime}\rVert$.

###  5.1  $\mathit{Cl}_{0,2}$ algebra

In this algebra according to Table 1 the norm (or magnitude) of $B^{\prime}$
is $\lvert B^{\prime}\rvert=\sqrt{-B^{\prime
2}}=\sqrt{B^{\prime}\widetilde{\wideparen{B^{\prime}}}}=\sqrt{b_{1}^{2}+b_{2}^{2}+b_{12}^{2}}$.
The logarithm defining equation is $\log\mathsf{B}=\mathsf{A}$, where
$\mathsf{B}$ is a given MV, and coefficients of $\mathsf{A}$ are to be
determined. Since $B^{\prime 2}<0$ the exponential of logarithm can be
expanded by trigonometric functions,

$\mathrm{e}^{\log\mathsf{B}}=\mathrm{e}^{\mathsf{A}}=\mathrm{e}^{a_{0}+A^{\prime}}=\mathrm{e}^{a_{0}}\big{(}\cos\lvert
A^{\prime}\rvert+\frac{A^{\prime}}{\lvert A^{\prime}\rvert}\sin\lvert
A^{\prime}\rvert\big{)},$ (5.1)

from which we write the following relation between ’vectors’ $B^{\prime}$ and
$A^{\prime}$,

$b_{0}+B^{\prime}=\mathrm{e}^{a_{0}}\big{(}\cos\lvert
A^{\prime}\rvert+\frac{A^{\prime}}{\lvert A^{\prime}\rvert}\sin\lvert
A^{\prime}\rvert\big{)}.$ (5.2)

Equation (5.2) can be rewritten as a system of two equations,

$b_{0}=\mathrm{e}^{a_{0}}\cos\lvert A^{\prime}\rvert,\quad
B^{\prime}=\mathrm{e}^{a_{0}}\frac{A^{\prime}}{\lvert
A^{\prime}\rvert}\sin\lvert A^{\prime}\rvert,$ (5.3)

where the second equation, in fact, represents three scalar equations. The
system (5.3) can be solved with respect to $a_{0}$ and $A^{\prime}$ in the
following way. After squaring both sides of (5.3) and noting that in
$\mathit{Cl}_{0,2}$ $B^{\prime 2}=-\lvert B^{\prime}\rvert^{2}$, we have

$b_{0}^{2}=\mathrm{e}^{2a_{0}}\cos^{2}\lvert A^{\prime}\rvert,\quad\lvert
B^{\prime}\rvert^{2}=\mathrm{e}^{2a_{0}}\sin^{2}\lvert A^{\prime}\rvert.$
(5.4)

The sum gives $b_{0}^{2}+\lvert B^{\prime}\rvert^{2}=\mathrm{e}^{2a_{0}}$ from
which and $b_{0}^{2}+\lvert B^{\prime}\rvert^{2}=\lvert\mathsf{B}\rvert^{2}$
follows

$a_{0}=\log\lvert\mathsf{B}\rvert.$ (5.5)

The ratio of equations in (5.4) gives $\lvert
B^{\prime}\rvert/b_{0}=\tan\lvert A^{\prime}\rvert$.The inverse of the latter
is

$\lvert A^{\prime}\rvert=\text{arctan}\big{(}\lvert
B^{\prime}\rvert/b_{0}\big{)}.$ (5.6)

To express the ‘vector’ $A^{\prime}$ in terms of $B^{\prime}$, the second
equation in (5.3) is divided by the first,

$\frac{B^{\prime}}{b_{0}}=\frac{A^{\prime}}{\lvert A^{\prime}\rvert}\tan\lvert
A^{\prime}\rvert.$ (5.7)

As follows from!(5.6) $\tan\lvert A^{\prime}\rvert=\lvert
B^{\prime}\rvert/b_{0}$, therefore, the Eq. (5.7) reduces to

$\frac{B^{\prime}}{b_{0}}=\frac{A^{\prime}}{\lvert
A^{\prime}\rvert}\frac{\lvert B^{\prime}\rvert}{b_{0}},$ (5.8)

from which the property $B^{\prime}/\lvert B^{\prime}\rvert=A^{\prime}/\lvert
A^{\prime}\rvert$, i.e. $B^{\prime}$ and $A^{\prime}$ are parallel in
$\\{\mathrm{i},\mathrm{j},\mathrm{k}\\}$ space, follows. The latter along with
(5.5) allow to get

$\log\mathsf{B}=a_{0}+A^{\prime}=\log\lvert\mathsf{B}\rvert+\lvert
A^{\prime}\rvert\frac{B^{\prime}}{\lvert B^{\prime}\rvert}.$ (5.9)

Finally, the needed generic logarithm formula is

$\mathit{Cl}_{0,2}:\
\log\mathsf{B}=\log\lvert\mathsf{B}\rvert+\frac{B^{\prime}}{\lvert
B^{\prime}\rvert}\left(\text{arctan}\frac{\lvert
B^{\prime}\rvert}{b_{0}}\right),\quad B^{\prime 2}<0.$ (5.10)

To logarithm (5.10) we may add a free MV $\mathsf{F}=2\pi k\hat{F}^{\prime}$,
where $\hat{F}^{\prime}$ plays the role of imaginary unit,
$(\hat{F}^{\prime})^{2}=-1$. In addition, it satisfies $\exp(2\pi
k\hat{F}^{\prime})=1$ and $\lvert\hat{F}^{\prime}\rvert=1$. As we shall see
the free MV takes into account the multi-valuedness of arc tangent. Then,
$\text{log}{\mathsf{B}}=\mathsf{A}+\mathsf{F}=\mathsf{A}+2\pi
k\hat{F}^{\prime}$. Since
$\mathsf{F}=f_{0}+f_{1}\mathbf{e}_{1}+f_{2}\mathbf{e}_{2}+f_{12}\mathbf{e}_{12}=f_{0}+F^{\prime}$
and $\lvert F^{\prime}\rvert=\sqrt{-F^{\prime
2}}=\big{(}F^{\prime}\widetilde{\wideparen{F^{\prime}}}\big{)}^{1/2}=\sqrt{f_{1}^{2}+f_{2}^{2}+f_{12}^{2}}$,
we have

$\mathrm{e}^{\mathsf{F}}=\mathrm{e}^{f_{0}}\big{(}\cos\lvert
F^{\prime}\rvert+\frac{F^{\prime}}{\lvert F^{\prime}\rvert}\sin\lvert
F^{\prime}\rvert\big{)}=1,$ (5.11)

which is satisfied if $f_{0}=0$ and $\lvert
F^{\prime}\rvert=\sqrt{f_{1}^{2}+f_{2}^{2}+f_{12}^{2}}=2\pi k,\quad
k\in\mathbb{Z}.$ Thus, in $\mathit{Cl}_{0,2}$ we have that the free MV is
$\mathsf{F}=2\pi k\hat{F}^{\prime}=2\pi
k\frac{f_{1}\mathbf{e}_{1}+f_{2}\mathbf{e}_{2}+f_{12}\mathbf{e}_{12}}{\sqrt{f_{1}^{2}+f_{2}^{2}+f_{12}^{2}}}=2\pi
k\tfrac{F^{\prime}}{\lvert F^{\prime}\rvert}$ that represents all possible
‘vectors’ the ends of which lie on a sphere of radius equal to $1$ in the 3D
anti-Euclidean space $\\{\mathrm{i},\mathrm{j},\mathrm{k}\\}$. When $k=0$ then
$\lvert F^{\prime}\rvert=0$ and $\mathsf{F}=0$, because $f_{0}=0$. Thus, we
conclude that the generic solution of equation
$\text{log}{\mathsf{B}}=\mathsf{A}$ represents the principal value of argument
$\varphi=\arctan\big{(}\lvert
B^{\prime}\rvert/b_{0}\big{)}=\arctan\big{(}\sqrt{b_{1}^{2}+b_{2}^{2}+b_{3}^{2}}/b_{0}\big{)}$
in the range $0\leq\varphi<2\pi$ if $k=0$. When multi-valuedness is included
the Hamilton quaternion logarithm takes the form

$\log\mathsf{B}+\mathsf{F}=\log\lvert\mathsf{B}\rvert+\frac{B^{\prime}}{\lvert
B^{\prime}\rvert}\Big{(}\arctan\frac{\lvert B^{\prime}\rvert}{b_{0}}+2\pi
k\Big{)},\quad B^{\prime 2}<0,\quad k\in\mathbb{Z}.$ (5.12)

$\lvert\mathsf{B}\rvert=(b_{0}^{2}+\lvert B^{\prime}\rvert^{2})^{1/2}$. The
formula (5.12) satisfies
$\mathrm{e}^{\text{log}\mathsf{B}+\mathsf{F}}=\mathsf{B}$ for all integers
$k$. Thus, in $\mathit{Cl}_{0,2}$ the free MV is
$\mathsf{F}=F^{\prime}=(B^{\prime}/\lvert B^{\prime}\rvert)2\pi k$, where the
‘vector’ $(B^{\prime}/\lvert B^{\prime}\rvert)$ plays the role of an imaginary
unit (compare with Eq. (3.3)). When $k=0$ we return back to the principal
logarithm. Thus, after replacement of the arc tangent by a double-argument arc
tangent in order to take account of all four quadrants correctly, the generic
formula (5.12) with special cases included becomes:

$\displaystyle\mathit{Cl}_{0,2}:\quad\log(\mathsf{B})=$
$\displaystyle\begin{cases}\log\lvert\mathsf{B}\rvert+\Bigl{(}\arctan\bigl{(}b_{0},\lvert
B^{\prime}\rvert\bigr{)}+2\pi k\Bigr{)}\frac{B^{\prime}}{\lvert
B^{\prime}\rvert},\quad\lvert B^{\prime}\rvert\neq 0,\\\ \log(b_{0})+2\pi
k\,\hat{F}^{\prime},\quad(\lvert B^{\prime}\rvert=0)\land(b_{0}>0),\\\
\log(-b_{0})+\pi(2k+1)\,\hat{F}^{\prime},\quad(\lvert
B^{\prime}\rvert=0)\land(b_{0}<0).\end{cases}$ (5.13)

To summarize, we have shown that, similar to complex number logarithm, the
Hamilton number logarithm is a multi-valued function too.

Example 4. $\mathit{Cl}_{0,2}$:
$\mathsf{B}=2+4\mathbf{e}_{1}-5\mathbf{e}_{2}-\mathbf{e}_{12}$.
$\lvert\mathsf{B}\rvert=\sqrt{46}$, $B^{\prime 2}=-42<0$, $\lvert
B^{\prime}\rvert=\sqrt{42}$. The principal logarithm is
$\log\mathsf{B}=\log\sqrt{46}+\left(\text{arctan}\frac{\sqrt{42}}{2}\right)\frac{4\mathbf{e}_{1}-5\mathbf{e}_{2}-\mathbf{e}_{12}}{\sqrt{42}}$
$\approx 1.914+4.662\mathbf{e}_{1}-5.826\mathbf{e}_{2}-1.165\mathbf{e}_{12}$ .
After exponentiation $\mathrm{e}^{\text{log}\mathsf{B}}$ we recover the
initial MV $\mathsf{B}$.

###  5.2 $\mathit{Cl}_{1,1}$ and $\mathit{Cl}_{2,0}$ algebras, when $B^{\prime
2}\leq 0$

When $B^{\prime 2}=-B^{\prime}\widetilde{\wideparen{B^{\prime}}}<0$, the
exponentials for both $\mathit{Cl}_{1,1}$ and $\mathit{Cl}_{2,0}$ a;algebra
are expressed in trigonometric functions in the same way as for
$\mathit{Cl}_{0,2}$ but the norm replaced by semi-norm (see Table 1). The free
MV $\mathsf{F}=f_{0}+\hat{F}^{\prime}$ also satisfies the condition
$\mathrm{e}^{2\pi k\mathsf{F}}=1$ from which we have $f_{0}=0$ and

$\begin{split}&\mathit{Cl}_{1,1}:\quad\hat{F}^{\prime}=f_{1}\mathbf{e}_{1}+f_{12}\mathbf{e}_{12}+\sqrt{1+f_{1}^{2}+f_{12}^{2}}\
\mathbf{e}_{2},\\\
&\mathit{Cl}_{2,0}:\quad\hat{F}^{\prime}=f_{1}\mathbf{e}_{1}+f_{2}\mathbf{e}_{2}-\sqrt{1+f_{1}^{2}+f_{2}^{2}}\
\mathbf{e}_{12},\end{split}$ (5.14)

with properties $\lVert\hat{F}^{\prime}\rVert=1$ and $\hat{F}^{\prime 2}=-1$.

Generic logarithm for both $\mathit{Cl}_{1,1}$ and $\mathit{Cl}_{2,0}$ are
given by equations similar to (5.13) but with $\hat{F}^{\prime}$ replaced by
(LABEL:hatFCL11CL29) and norm replaced by semi-norm,

$\displaystyle\mathit{Cl}_{1,1},\mathit{Cl}_{2,0}:\ \log(\mathsf{B})=$
$\displaystyle\begin{cases}\log(\lVert\mathsf{B}\rVert)+\bigl{(}\arctan(b_{0},\lVert
B^{\prime}\rVert)+2\pi k\bigr{)}\frac{B^{\prime}}{\lVert
B^{\prime}\rVert},\quad(B^{\prime 2}<0),\\\
\log(b_{0})+B^{\prime}/b_{0},\quad(B^{\prime 2}=0)\land(b_{0}>0),\\\
\log(-b_{0})+\pi(2k+1)\,\hat{F}^{\prime},\quad(B^{\prime}=0)\land(b_{0}<0).\end{cases}$
(5.15)

The last two equations represent special cases. Now the logarithm of MV exists
when $\lVert B^{\prime}\rVert\neq 0$ and $\lVert B^{\prime}\rVert=0$. More
specific cases are presented in Subsec. 5.4.

Example 5. $\mathit{Cl}_{1,1}$:
$\mathsf{B}=2+4\mathbf{e}_{1}-5\mathbf{e}_{2}-\mathbf{e}_{12}$, $B^{\prime
2}=-8$, $\lVert\mathsf{B}\rVert=\sqrt{12}$, $\lVert
B^{\prime}\rVert=\sqrt{8}$. The answer
$\log\mathsf{B}=\tfrac{1}{2}\log(12)+\tfrac{1}{2\sqrt{2}}\left(\text{arctan}\sqrt{2}+2\pi
k\right)(4\mathbf{e}_{1}-5\mathbf{e}_{2}-\mathbf{e}_{12})$ satisfies
$\mathrm{e}^{\log\mathsf{B}}=\mathsf{B}$.

Example 6. $\mathit{Cl}_{2,0}$,
$\mathsf{B}=2+5\mathbf{e}_{1}-4\mathbf{e}_{2}-7\mathbf{e}_{12}$ $B^{\prime
2}=-8$. $\lVert B^{\prime}\rVert=\sqrt{8}$ $\lVert\mathsf{B}\rVert=\sqrt{12}$.
Answer:
$\log\mathsf{B}=\frac{\log(12)}{2}+\tfrac{1}{2\sqrt{2}}(\text{arctan}(\sqrt{2})+2\pi
k)(5\mathbf{e}_{1}-4\mathbf{e}_{2}-7\mathbf{e}_{12})$. The answer satisfies
$\mathrm{e}^{\log\mathsf{B}}=\mathsf{B}$.

Example 7. $\mathit{Cl}_{2,0}$.
$\mathsf{B}=2+3\mathbf{e}_{1}-4\mathbf{e}_{2}-5\mathbf{e}_{12}$; $B^{\prime
2}=0$, $\lVert B^{\prime}\rVert=0$, $\lVert\mathsf{B}\rVert=2$.
Answer:
$\log(\mathsf{B})=\log(b_{0})+B^{\prime}/b_{0}=\log(2)+\frac{1}{2}(3\mathbf{e}_{1}-4\mathbf{e}_{2}-5\mathbf{e}_{12})$.
$\mathrm{e}^{\log\mathsf{B}}=\mathsf{B}$..

Example 8. $\mathit{Cl}_{2,0}$. $\mathsf{B}=-2$,$B^{\prime 2}=0$, $\lVert
B^{\prime}\rVert=0$.
$\log\mathsf{B}=\log(-b_{0})+\pi(2k+1)\hat{F}^{\prime}=\log
2+\pi(2k+1)(f_{1}\mathbf{e}_{1}+f_{2}\mathbf{e}_{2}-\sqrt{1+f_{1}^{2}+f_{2}^{2}})\mathbf{e}_{12}$.
The answer satisfies $\mathrm{e}^{\log\mathsf{B}}=-2$.

###  5.3 $\mathit{Cl}_{1,1}$ and $\mathit{Cl}_{2,0}$ algebras, when $B^{\prime
2}>0$

When $B^{\prime 2}>0$, calculations proceed in a similar way. Therefore, only
intermediate results are put down briefly. Let the general MV be
$\mathsf{B}=b_{0}+B^{\prime}$. The semi-norm in $\mathit{Cl}_{1,1}$ is $\lVert
B^{\prime}\rVert=\sqrt{b_{1}^{2}-b_{2}^{2}+b_{12}^{2}}$ and in
$\mathit{Cl}_{2,0}$ it is $\lVert
B^{\prime}\rVert=\sqrt{b_{1}^{2}+b_{2}^{2}-b_{12}^{2}}0$, where expression
under the root should be positive. As in previous case the defining equation
is $\log{\mathsf{B}}=\mathsf{A}$, where $\mathsf{A}=a_{0}+A^{\prime}$. Since
the square of $A^{\prime}$ now is positive scalar, $A^{{}^{\prime}2}>0$, in
agreement with the Table 1, the exponential is expanded in hyperbolic
functions,

$\mathrm{e}^{\log\mathsf{B}}=\mathrm{e}^{a_{0}+A^{\prime}}=\mathrm{e}^{a_{0}}\big{(}\cosh\lVert
A^{\prime}\rVert+\frac{A^{\prime}}{\lVert A^{\prime}\rVert}\sinh\lVert
A^{\prime}\rVert\big{)}.$ (5.16)

Thus, we have the following relation between ‘vectors’ $B^{\prime}$ and
$A^{\prime}$,

$b_{0}+B^{\prime}=\mathrm{e}^{a_{0}}\big{(}\cosh\lVert
A^{\prime}\rVert+\frac{A^{\prime}}{\lVert A^{\prime}\rVert}\sinh\lVert
A^{\prime}\rVert\big{)},$ (5.17)

that may be rewritten as a system of equations

$b_{0}=\mathrm{e}^{a_{0}}\cosh\lVert A^{\prime}\rVert,\quad
B^{\prime}=\mathrm{e}^{a_{0}}\frac{A^{\prime}}{\lVert
A^{\prime}\rVert}\sinh\lVert A^{\prime}\rVert.$ (5.18)

Squaring of Eqs. (5.18) and the property $B^{\prime 2}=\lVert
B^{\prime}\rVert^{2}$ gives

$b_{0}^{2}=\mathrm{e}^{2a_{0}}\cosh^{2}\lVert A^{\prime}\rVert,\quad\lVert
B^{\prime}\rVert^{2}=\mathrm{e}^{2a_{0}}\sinh^{2}\lVert A^{\prime}\rVert.$
(5.19)

Now, applying the property $\cosh^{2}{A^{\prime}}-\sinh^{2}{A^{\prime}}=1$,
the difference of equations in (5.19) yields the scalar equation
$b_{0}^{2}-\lVert B^{\prime}\rVert^{2}=\mathrm{e}^{2a_{0}}$, from which and
the relation $\lVert\mathsf{B}\rVert^{2}=b_{0}^{2}-\lVert
B^{\prime}\rVert^{2}$, follows $a_{0}=\log\lVert\mathsf{B}\rVert$. Also, from
Eq (5.19) we have the ratio $\lVert B^{\prime}\rVert/b_{0}=\tanh\lVert
A^{\prime}\rVert$, from which we find equation analogous Eq (5.6),
$\lVert{A^{\prime}\lVert=\text{artanh}\big{(}\lVert
B^{\prime}}\rVert/b_{0}\big{)}$, where artanh is the area hyperbolic tangent.
To express $A^{\prime}$ in terms of $B^{\prime}$, we divide equations in
(5.18),

$\frac{B^{\prime}}{b_{0}}=\frac{A^{\prime}}{\lVert
A^{\prime}\rVert}\tanh\lVert A^{\prime}\rVert.$ (5.20)

Since $\tanh\lVert A^{\prime}\rVert=\lVert B^{\prime}\rVert/b_{0}$, the Eq.
(5.20) can be reduced to

$\frac{B^{\prime}}{b_{0}}=\frac{A^{\prime}}{\lVert
A^{\prime}\rVert}\frac{\lvert B^{\prime}\rvert}{b_{0}}.$ (5.21)

from which the property $B^{\prime}/\lVert B^{\prime}\rVert=A^{\prime}/\lVert
A^{\prime}\rVert$ follows. The latter allows to get the required formula for
the principal logarithm,

$\mathit{Cl}_{1,1},\mathit{Cl}_{2,0}:\
\log\mathsf{B}=\log\lVert\mathsf{B}\rVert+\left(\text{artanh}\frac{\lVert
B^{\prime}\rVert}{b_{0}}\right)\frac{B^{\prime}}{\lVert
B^{\prime}\rVert},\quad(B^{\prime 2}>0)\land(b_{0}\neq 0).$ (5.22)

To this formula we should add a free MV
$\mathsf{F}=f_{0}+F^{\prime}=f_{0}+f_{1}\mathbf{e}_{1}+f_{2}\mathbf{e}_{2}+f_{12}\mathbf{e}_{12}$
that satisfies $\mathrm{e}^{\mathsf{F}}=1$, or

$\mathrm{e}^{\mathsf{F}}=\mathrm{e}^{f_{0}}\mathrm{e}^{F^{\prime}}=\mathrm{e}^{f_{0}}\bigl{(}\cosh\lVert
F^{\prime}\rVert+\frac{F^{\prime}}{\lVert F^{\prime}\rVert}\sinh\lVert
F^{\prime}\rVert\bigr{)}=1.$ (5.23)

The solution of this MV equation (equivalently of four scalar equations) is
$f_{0}=0$, and $\lVert
F^{\prime}\rVert=\sqrt{f_{1}^{2}-f_{2}^{2}+f_{12}^{2}}=0$ for
$\mathit{Cl}_{1,1}$ and $\lVert
F^{\prime}\rVert=\sqrt{f_{1}^{2}+f_{2}^{2}-f_{12}^{2}}=0$ for
$\mathit{Cl}_{2,0}$. From this we conclude that $\mathsf{F^{\prime}}=0$. Thus,
in the case $B^{2}>0$ the principal logarithm is the only solution.

###  5.4 $\mathit{Cl}_{1,1}$ and $\mathit{Cl}_{2,0}$ algebras: summary

Taking into account the generic and special cases finally we can write

$\displaystyle\mathit{Cl}_{1,1},\mathit{Cl}_{2,0}:\ \log(\mathsf{B})=$
$\displaystyle\begin{cases}\log\lVert\mathsf{B}\rVert+\bigl{(}\arctan(b_{0},\lVert
B^{\prime}\rVert)+2\pi k\bigr{)}\frac{B^{\prime}}{\lVert
B^{\prime}\rVert},\qquad(B^{\prime 2}<0),\\\
\log\lVert\mathsf{B}\rVert+\operatorname{arctanh}\bigl{(}\frac{\lVert
B^{\prime}\rVert}{b_{0}}\bigr{)}\frac{B^{\prime}}{\lVert
B^{\prime}\rVert},\qquad(B^{\prime
2}>0)\land(b_{0}>0)\land(b_{0}^{2}-B^{\prime 2}>0),\\\
\log(b_{0})+\frac{1}{2}\log(0_{+})\bigl{(}1-\frac{B^{\prime}}{\lVert
B^{\prime}\rVert}\bigr{)}+\frac{1}{2}\log(2)\bigl{(}1+\frac{B^{\prime}}{\lVert
B^{\prime}\rVert}\bigr{)},(B^{\prime
2}>0)\land(b_{0}>0)\land(\lVert\mathsf{B}\lVert=0),\\\
\log(b_{0})+\frac{B^{\prime}}{b_{0}},\quad(B^{\prime 2}=0)\land(b_{0}>0),\\\
\log(-b_{0})+(\pi+2\pi k)\hat{F}^{\prime},\quad(B^{\prime}=0)\land(b_{0}\leq
0),\\\ \varnothing,\quad\textrm{no solution},\quad(B^{\prime
2}>0)\land\bigl{(}(b_{0}\leq 0)\lor(b_{0}^{2}-B^{\prime 2}\leq
0)\bigl{)}.\end{cases}$ (5.24)

where $\lVert\mathsf{B}\rVert=\sqrt{\operatorname{abs}(b_{0}^{2}-B^{\prime
2})}$ and $\lVert B^{\prime}\rVert=\sqrt{\operatorname{abs}(B^{\prime 2})}$.
Explicit form of a free MV $\hat{F}^{\prime}$ is algebra dependent and is
given in (LABEL:hatFCL11CL29).

In conclusion, we shall remark that in $\mathit{Cl}_{0,2}$ algebra the GA
logarithm is defined for all MVs. However, in the remaining 2D algebras we
have to satisfy the conditions for coefficients for a logarithm to exist.
Thus, in these algebras there are sectors in a domain of argument where
logarithm does not exist at all.

Example 9. $\mathit{Cl}_{2,0}$, line 2 in (5.4). Case $(B^{\prime
2}>0)\land(b_{0}>0)\land(b_{0}^{2}-B^{\prime 2}>0)$.
$\mathsf{B}=2+B^{\prime}=2+5\mathbf{e}_{1}-\mathbf{e}_{2}-5\mathbf{e}_{12}$;
$B^{\prime 2}=1$, $\lVert\mathsf{B}\rVert=\sqrt{3}$, $\lVert
B^{\prime}\rVert=1$, $\lVert\mathsf{B}\rVert^{2}=b_{0}^{2}-B^{\prime 2}=3$.
Answer:
$\log(\mathsf{B})=\frac{\log(3)}{2}+\operatorname{arctanh}\left(\frac{1}{2}\right)(5\mathbf{e}_{1}-5\mathbf{e}_{12}-\mathbf{e}_{2})$.

Example 10. $\mathit{Cl}_{2,0}$, line 3 in (5.4). Case $(B^{\prime
2}>0)\land(b_{0}>0)\land(b_{0}^{2}-B^{\prime 2}=0)$.
$\mathsf{B}=9-9\mathbf{e}_{1}+8\mathbf{e}_{2}+8\mathbf{e}_{12}$; $B^{\prime
2}=81$, $\lVert B^{\prime}\rVert=9$,
$\lVert\mathsf{B}\rVert^{2}=b_{0}^{2}-B^{\prime 2}=0$.
Answer:
$\log(\mathsf{B})=\frac{1}{18}\log\left(\frac{2}{0_{+}}\right)(-9\mathbf{e}_{1}+8\mathbf{e}_{2}+8\mathbf{e}_{12})+\frac{1}{2}\log(2+0_{+})+\log(9)$,
where $0_{+}$ is infinitesimally small positive number. $\lim_{0_{+}\to
0}\mathrm{e}^{\log\mathsf{B}}=\mathsf{B}$

Example 11 $\mathit{Cl}_{2,0}$, line 4 in (5.4). Case $(\mathsf{B}^{\prime
2}=0)\land(b_{0}>0)$
$\mathsf{B}=2+3\mathbf{e}_{1}-4\mathbf{e}_{2}-5\mathbf{e}_{12}=2+B^{\prime}$,
$\lVert\mathsf{B}\rVert=2$, $B^{\prime 2}=0$, $\lVert B^{\prime}\rVert=0$.
Answer: $\log\mathsf{B}=\log 2+B^{\prime}/b_{0}=\log
2+\frac{1}{2}(3\mathbf{e}_{1}-4\mathbf{e}_{2}-5\mathbf{e}_{12})$.

Example 12. $\mathit{Cl}_{2,0}$, line 5 in (5.4). Case
$(\mathsf{B}^{\prime}=0)\land(b_{0}\leq 0)$. The logarithm of MV
$\mathsf{B}=-2$ is $\log(\mathsf{B})=\log(2)+(\pi+2\pi
k)\left(f_{1}\mathbf{e}_{1}-\mathbf{e}_{12}\sqrt{1+f_{1}^{2}+f_{2}^{2}}+f_{2}\mathbf{e}_{2}\right)$.
After exponentiation $\hat{F}^{\prime}$ simplifies out and we get
$\mathrm{e}^{\log\mathsf{B}}=-2$,

Example 13. $\mathit{Cl}_{1,1}$, line 6 in (5.4)
$\mathsf{B}=2+5\mathbf{e}_{1}-4\mathbf{e}_{2}-7\mathbf{e}_{12};$ $B^{\prime
2}=58$, $b_{0}^{2}-B^{\prime 2}=-54$. Logarithm does not exist, since under
the condition $B^{\prime 2}>0$ solution exists only when $b_{0}^{2}-B^{\prime
2}>0$ (case 2 of (5.4)). Here we have $b_{0}^{2}-B^{\prime 2}<0$ and,
therefore by line 6 of (5.4) the solution set is empty.

The knowledge of logarithm and exponential provides a possibility to calculate
the square root of a MV by formula
$\sqrt{\mathsf{B}}=\pm\exp(\frac{1}{2}\log\mathsf{B})$. For example, for
$\mathsf{B}=2-\mathbf{e}_{1}+2\mathbf{e}_{12}$ in $\mathit{Cl}_{2,0}$ we have
$B^{\prime 2}=-3$ and $\lVert\mathsf{B}\rVert=\sqrt{7}$. Then, using the
formula (5.15) one finds $\log\mathsf{B}=\frac{1}{2}\log
7-\tfrac{1}{\sqrt{3}}(\mathbf{e}_{1}-2\mathbf{e}_{12})\arctan\tfrac{1}{\sqrt{3}}$
and after multiplication by $\tfrac{1}{2}$ and application of exponential one
obtains the root
$\sqrt{\mathsf{B}}=\tfrac{2+\sqrt{7}-\mathbf{e}_{1}+2\mathbf{e}_{12}}{\sqrt{2(2+\sqrt{7})}}$.
In this way calculated root coincides with a root formula (6.3) in the next
subsection. It is easy to verify that the geometric product
$\sqrt{\mathsf{B}}\sqrt{\mathsf{B}}$ simplifies to the initial $\mathsf{B}$.
The formula $\sqrt{\mathsf{B}}=\pm\exp(\frac{1}{2}\log\mathsf{B})$ gives only
two (plus/minus) roots. If $\mathsf{B}$ is a unity, $\mathsf{B}=1$, then the
square root exists for all algebras since
$\sqrt{1}=\pm\exp(\tfrac{1}{2}\log(1))=\pm 1$. From works [17, 16] we know
that in the Clifford number algebra, in general, the square root of MV is a
multi-valued function. Below in Subsec. 6 such (isolated) roots are presented
for $\sqrt{+1}$ and $\sqrt{-1}$. The multiple roots also have been found for a
general MV in $\mathit{Cl}_{1,1}$ (see the next subsection) where up to four
roots may happen simultaneously. In higher dimensional GAs the number of roots
may be even larger [17].

##  6 Addendum: Formulas for square roots of MV

Below the square roots $\sqrt{\mathsf{B}}$ for 1D and 2D algebras that may be
useful in practice are presented. They were calculated from the defining
equation $\mathsf{B}=\mathsf{A}^{2}$ which is equivalent to a system of two in
2D or four 2D real coupled equations.

For $\mathit{Cl}_{0,1}$ there are two (plus and minus) roots of
$\mathsf{B}=b_{0}+\mathbf{e}_{1}b_{1}$,

$\mathit{Cl}_{0,1}:\sqrt{\mathsf{B}}=\pm\frac{\sqrt{b_{0}^{2}+b_{1}^{3}}+(b_{0}+b_{1}\mathbf{e}_{1})}{\sqrt{2}\sqrt{b_{0}+\sqrt{b_{0}^{2}+b_{1}^{2}}}}=\pm\frac{\lvert\mathsf{B}\rvert\mathsf{B}}{\sqrt{2}\sqrt{\langle\mathsf{B}\rangle_{0}+\lvert\mathsf{B}\rvert}},$
(6.1)

where $\lvert\mathsf{B}\rvert=\sqrt{\mathsf{B}\wideparen{\mathsf{B}}}$ is the
magnitude (norm) and $\langle\mathsf{B}\rangle_{0}=b_{0}$ is the scalar part
of MV. The roots (6.1) exists for all MVs $\mathsf{B}\neq 0$.

For $\mathit{Cl}_{1,0}$ algebra, in general, there are four roots of
$\mathsf{B}=b_{0}+\mathbf{e}_{1}b_{1}$,

$\mathit{Cl}_{1,0}:\sqrt{\mathsf{B}}=\left\\{\pm\frac{-\sqrt{b_{0}^{2}-b_{1}^{2}}+b_{0}+b_{1}\mathbf{e}_{1}}{\sqrt{2}\sqrt{b_{0}-\sqrt{b_{0}^{2}-b_{1}^{2}}}},\
\pm\frac{\sqrt{b_{0}^{2}-b_{1}^{2}}+b_{0}+b_{1}\mathbf{e}_{1}}{\sqrt{2}\sqrt{b_{0}+\sqrt{b_{0}^{2}-b_{1}^{2}}}}\right\\}.$
(6.2)

If $b_{0}>b_{1}$ all roots are different. If $b_{0}=b_{1}$ only two distinct
roots remain.

In 2D algebras the MV is
$\mathsf{B}=b_{0}+b_{1}\mathbf{e}_{1}+b_{2}\mathbf{e}_{2}+b_{12}\mathbf{e}_{12}$.
The square root has the same form for all algebras:

$\mathit{Cl}_{0,2},\mathit{Cl}_{1,1},\mathit{Cl}_{2,0}:\sqrt{\mathsf{B}}=\pm\frac{b_{0}+\sqrt{\operatorname{Det}\mathsf{B}}+b_{1}\mathbf{e}_{1}+b_{2}\mathbf{e}_{2}+b_{12}\mathbf{e}_{12}}{\sqrt{2}\sqrt{b_{0}+\sqrt{\operatorname{Det}\mathsf{B}}}},$
(6.3)

For individual algebras the determinant is

$\operatorname{Det}\mathsf{B}=\mathsf{B}\widetilde{\wideparen{\mathsf{B}}}=\begin{cases}&b_{0}^{2}+b_{1}^{2}+b_{2}^{2}+b_{12}^{2}\quad\text{for}\
\mathit{Cl}_{0,2},\\\
&b_{0}^{2}-b_{1}^{2}+b_{2}^{2}-b_{12}^{2}\quad\text{for}\
\mathit{Cl}_{1,1},\\\
&b_{0}^{2}-b_{1}^{2}-b_{2}^{2}+b_{12}^{2}\quad\text{for}\
\mathit{Cl}_{2,0}.\\\ \end{cases}$ (6.4)

For Hamilton quaternion the square root exists for all MVs. For
$\mathit{Cl}_{1,1}$ and $\mathit{Cl}_{2,0}$ algebras the roots exist when the
determinant is positive or zero. In $\mathit{Cl}_{1,1}$ an additional
plus/minus roots may appear if $\operatorname{Det}\mathsf{B}>b_{0}>0$ [12],

$\sqrt{\mathsf{B}}=\pm\frac{-b_{0}+\sqrt{\operatorname{Det}\mathsf{B}}-(b_{1}\mathbf{e}_{1}+b_{2}\mathbf{e}_{2}+b_{12}\mathbf{e}_{12})}{\sqrt{2}\sqrt{-b_{0}-\sqrt{\operatorname{Det}\mathsf{B}}}}.$
(6.5)

Square roots of $+1$ in 2D. The square roots of $\mathsf{B}=+1$ in 2D are

$\displaystyle\sqrt{+1}=\begin{cases}\mathit{Cl}_{0,2}:\ \\{\pm 1\\},\\\
\mathit{Cl}_{1,1}:\ \\{\pm
1,\mathbf{e}_{1},\tfrac{1}{\sqrt{2}}(\mathbf{e}_{1}\pm\mathbf{e}_{12})\\}\textrm{
and
}\pm(c_{1}\mathbf{e}_{1}+c_{2}\mathbf{e}_{2}\pm\sqrt{1+c_{1}^{2}+c_{2}^{2}}\,\mathbf{e}_{12}),\\\
\mathit{Cl}_{2,0}:\ \\{\pm
1,\mathbf{e}_{1},\mathbf{e}_{2},\tfrac{1}{\sqrt{2}}(\mathbf{e}_{1}\pm\mathbf{e}_{2})\\};\pm(c_{1}\mathbf{e}_{1}+c_{2}\mathbf{e}_{2}\pm\sqrt{-1+c_{1}^{2}+c_{2}^{2}}\,\mathbf{e}_{12}).\\\
\end{cases}$ (6.6)

In case of $\mathit{Cl}_{2,0}$ and $\mathit{Cl}_{1,1}$ the coefficients
$c_{1}$ and $c_{2}$ are arbitrary and may be considered as free parameters.
Their range is determined by expression under the square root which must be
positive. Thus, in this case we have two types of roots: the isolated roots
and the continuum of roots determined by free parameters $c_{i}$.

Square roots of $-1$ in 2D. The square roots of $\mathsf{B}=-1$ in 2D are

$\displaystyle\sqrt{-1}=\begin{cases}\mathit{Cl}_{0,2}:\\{\mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{12},\tfrac{1}{\sqrt{2}}(\mathbf{e}_{1}\pm\mathbf{e}_{2}),\tfrac{1}{\sqrt{2}}(\mathbf{e}_{1}\pm\mathbf{e}_{12}),\tfrac{1}{\sqrt{2}}(\mathbf{e}_{2}\pm\mathbf{e}_{12})\\}\\\
\qquad\qquad\qquad\qquad\text{ and
}\pm(c_{1}\mathbf{e}_{1}+c_{2}\mathbf{e}_{2}\pm\sqrt{1-c_{1}^{2}-c_{2}^{2}}\,\mathbf{e}_{12}),\\\
\mathit{Cl}_{1,1}:\\{\mathbf{e}_{2}\\}\text{ and
}\pm(c_{1}\mathbf{e}_{1}+c_{2}\mathbf{e}_{2}\pm\sqrt{-1-c_{1}^{2}+c_{2}^{2}}\,\mathbf{e}_{12}),\\\
\mathit{Cl}_{2,0}:\\{\mathbf{e}_{12}\\}\text{ and
}\pm(c_{1}\mathbf{e}_{1}+c_{2}\mathbf{e}_{2}\pm\sqrt{1+c_{1}^{2}+c_{2}^{2}}\,\mathbf{e}_{12}).\\\
\end{cases}$ (6.7)

##  7 Conclusions and discussion

The paper presents exponential and logarithm functions of multivector argument
for $\mathit{Cl}_{0,1}$, $\mathit{Cl}_{1,0}$ (1-dimensional commutative) and
$\mathit{Cl}_{0,2}$, $\mathit{Cl}_{1,1}$, $\mathit{Cl}_{2,0}$ (2-dimensional
non-commutative) Clifford geometric algebras (GAs). The well-known approach to
Hamilton quaternion identified by three imaginaries
$\\{\mathrm{i},\mathrm{j},\mathrm{k}\\}$ was generalized and adapted,
specifically, the imaginaries have been replaced by 2D unit multivectors the
squares of which are equal to $\pm 1$ and which have been constructed from
Clifford basis vectors $\\{\mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{12}\\}$.

The 2D basis-free exponentials in this approach assume a form of either
Euler’s or de Moivre’s rules. The principal logarithm was determined as an
inverse of respective exponential. $2\pi$ multiplicity in the logarithm was
included by adding a free MV $\mathsf{F}$ that satisfies the condition
$\mathrm{e}^{\mathsf{F}}=1$. The obtained formulas for exponential and
logarithm can be applied to quaternions too, since $\mathit{Cl}_{0,2}$,
$\mathit{Cl}_{1,1}$ and $\mathit{Cl}_{2,0}$ algebras are isomorphic to,
correspondingly, the Hamilton quaternion, coquaternion, and conectorine [13],

Since in GA the $n$-th root of MV is
$(\mathsf{A})^{1/n}=\exp\big{(}\tfrac{1}{n}\log(\mathsf{A})\big{)}$ the
obtained exponential and logarithm formulas may be applied to extract the
$n$-th root from general MV. However, this exp-log formula allows to calculate
no more then two (plus/minus) square roots. Workable formulas for square roots
are presented. In particular, using the defining equation we have found
explicit formulas for square roots in 1D and 2D Clifford algebras and the
sectors of their existence in MV coefficient space. In the space of MV
coefficients the sectors where the roots do not exist the logarithm does not
exist as well. Also, multiple square roots (up to four) have been found in
$\mathit{Cl}_{1,0}$ and $\mathit{Cl}_{1,1}$ algebras.

The presented results may be useful in applied GAs, especially in dealing with
GA differential equations the solutions of which are expressed through GA
exponentials [14, 23]. Finally, the low dimensional Clifford algebras may be
helpful in doing calculations in higher dimensional algebras as well, because
the former are subalgebras of the latter. Also, it is expected that the
described in the paper approach may be adapted to higher grade Clifford
algebras.

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