<s>
In	O
mathematics	O
,	O
a	O
function	O
of	O
a	O
motor	B-Algorithm
variable	I-Algorithm
is	O
a	O
function	O
with	O
arguments	O
and	O
values	O
in	O
the	O
split-complex	O
number	O
plane	O
,	O
much	O
as	O
functions	O
of	O
a	O
complex	O
variable	O
involve	O
ordinary	O
complex	O
numbers	O
.	O
</s>
<s>
Motor	B-Algorithm
variable	I-Algorithm
is	O
used	O
here	O
in	O
place	O
of	O
split-complex	O
variable	O
for	O
euphony	O
and	O
tradition	O
.	O
</s>
<s>
Functions	O
of	O
a	O
motor	B-Algorithm
variable	I-Algorithm
provide	O
a	O
context	O
to	O
extend	O
real	O
analysis	O
and	O
provide	O
compact	O
representation	O
of	O
mappings	O
of	O
the	O
plane	O
.	O
</s>
<s>
Nevertheless	O
,	O
some	O
of	O
the	O
aspects	O
of	O
conventional	O
complex	O
analysis	O
have	O
an	O
interpretation	O
given	O
with	O
motor	B-Algorithm
variables	I-Algorithm
,	O
and	O
more	O
generally	O
in	O
hypercomplex	O
analysis	O
.	O
</s>
<s>
The	O
following	O
exemplar	O
functions	O
f	O
have	O
domain	B-Algorithm
and	O
range	B-Algorithm
in	O
D	O
:	O
</s>
<s>
The	O
motor	B-Algorithm
variable	I-Algorithm
is	O
characterized	O
by	O
hyperbolic	O
angle	O
in	O
its	O
polar	O
coordinates	O
,	O
and	O
this	O
angle	O
is	O
preserved	O
by	O
motor	B-Algorithm
variable	I-Algorithm
linear	O
fractional	O
transformations	O
just	O
as	O
circular	O
angle	O
is	O
preserved	O
by	O
the	O
Möbius	O
transformations	O
of	O
the	O
ordinary	O
complex	O
plane	O
.	O
</s>
<s>
To	O
realize	O
the	O
linear	O
fractional	O
transformations	O
as	O
bijections	B-Algorithm
on	O
the	O
projective	O
line	O
a	O
compactification	O
of	O
D	O
is	O
used	O
.	O
</s>
<s>
In	O
the	O
theory	O
of	O
functions	O
of	O
a	O
motor	B-Algorithm
variable	I-Algorithm
special	O
attention	O
should	O
be	O
called	O
to	O
the	O
square	O
root	O
and	O
logarithm	O
functions	O
.	O
</s>
<s>
In	O
particular	O
,	O
the	O
plane	O
of	O
split-complex	O
numbers	O
consists	O
of	O
four	O
connected	O
components	O
and	O
the	O
set	O
of	O
singular	O
points	O
that	O
have	O
no	O
inverse	O
:	O
the	O
diagonals	O
z	O
=	O
x	O
±	O
x	O
j	O
,	O
x	O
∈	O
R	O
.	O
The	O
identity	O
component	O
,	O
namely	O
{	O
z	O
:	O
x	O
>	O
|y|	O
}	O
=	O
U1	O
,	O
is	O
the	O
range	B-Algorithm
of	O
the	O
squaring	O
function	O
and	O
the	O
exponential	O
.	O
</s>
<s>
Thus	O
it	O
is	O
the	O
domain	B-Algorithm
of	O
the	O
square	O
root	O
and	O
logarithm	O
functions	O
.	O
</s>
<s>
The	O
other	O
three	O
quadrants	O
do	O
not	O
belong	O
in	O
the	O
domain	B-Algorithm
because	O
square	O
root	O
and	O
logarithm	O
are	O
defined	O
as	O
one-to-one	B-Algorithm
inverses	O
of	O
the	O
squaring	O
function	O
and	O
the	O
exponential	O
function	O
.	O
</s>
<s>
The	O
Cauchy	O
–	O
Riemann	O
equations	O
that	O
characterize	O
holomorphic	O
functions	O
on	O
a	O
domain	B-Algorithm
in	O
the	O
complex	O
plane	O
have	O
an	O
analogue	O
for	O
functions	O
of	O
a	O
motor	B-Algorithm
variable	I-Algorithm
.	O
</s>
<s>
At	O
the	O
National	O
University	O
of	O
La	O
Plata	O
in	O
1935	O
,	O
J.C.	O
Vignaux	O
,	O
an	O
expert	O
in	O
convergence	O
of	O
infinite	O
series	O
,	O
contributed	O
four	O
articles	O
on	O
the	O
motor	B-Algorithm
variable	I-Algorithm
to	O
the	O
university	O
's	O
annual	O
periodical	O
.	O
</s>
<s>
He	O
then	O
proceeds	O
,	O
for	O
example	O
,	O
to	O
generalize	O
theorems	O
due	O
to	O
Cauchy	O
,	O
Abel	O
,	O
Mertens	O
,	O
and	O
Hardy	O
to	O
the	O
domain	B-Algorithm
of	O
the	O
motor	B-Algorithm
variable	I-Algorithm
.	O
</s>
<s>
Vignaux	O
completed	O
his	O
series	O
with	O
a	O
six-page	O
note	O
on	O
the	O
approximation	O
of	O
D-holomorphic	O
functions	O
in	O
a	O
unit	O
isotropic	O
rectangle	O
by	O
Bernstein	B-Algorithm
polynomials	I-Algorithm
.	O
</s>
<s>
The	O
multiplicative	O
inverse	O
function	O
is	O
so	O
important	O
that	O
extreme	O
measures	O
are	O
taken	O
to	O
include	O
it	O
in	O
the	O
mappings	O
of	O
differential	B-Language
geometry	I-Language
.	O
</s>
