<s>
In	O
mathematics	O
,	O
quaternionic	B-Algorithm
analysis	I-Algorithm
is	O
the	O
study	O
of	O
functions	O
with	O
quaternions	O
as	O
the	O
domain	B-Algorithm
and/or	O
range	O
.	O
</s>
<s>
Such	O
functions	O
can	O
be	O
called	O
functions	O
of	O
a	O
quaternion	B-Algorithm
variable	I-Algorithm
just	O
as	O
functions	O
of	O
a	O
real	O
variable	O
or	O
a	O
complex	O
variable	O
are	O
called	O
.	O
</s>
<s>
As	O
with	O
complex	O
and	O
real	O
analysis	O
,	O
it	O
is	O
possible	O
to	O
study	O
the	O
concepts	O
of	O
analyticity	B-Language
,	O
holomorphy	O
,	O
harmonicity	O
and	O
conformality	O
in	O
the	O
context	O
of	O
quaternions	O
.	O
</s>
<s>
The	O
projections	B-Algorithm
of	O
a	O
quaternion	O
onto	O
its	O
scalar	O
part	O
or	O
onto	O
its	O
vector	O
part	O
,	O
as	O
well	O
as	O
the	O
modulus	O
and	O
versor	O
functions	O
,	O
are	O
examples	O
that	O
are	O
basic	O
to	O
understanding	O
quaternion	O
structure	O
.	O
</s>
<s>
The	O
quaternion	O
multiplicative	O
inverse	O
is	O
another	O
fundamental	O
function	O
,	O
but	O
as	O
with	O
other	O
number	O
systems	O
,	O
and	O
related	O
problems	O
are	O
generally	O
excluded	O
due	O
to	O
the	O
nature	O
of	O
dividing	B-Algorithm
by	I-Algorithm
zero	I-Algorithm
.	O
</s>
<s>
Linear	B-Architecture
fractional	O
transformations	O
of	O
quaternions	O
can	O
be	O
represented	O
by	O
elements	O
of	O
the	O
matrix	O
ring	O
operating	O
on	O
the	O
projective	O
line	O
over	O
.	O
</s>
<s>
Quaternion	B-Algorithm
variable	I-Algorithm
theory	O
differs	O
in	O
some	O
respects	O
from	O
complex	O
variable	O
theory	O
.	O
</s>
<s>
Consequently	O
,	O
since	O
is	O
linear	B-Architecture
,	O
</s>
<s>
The	O
success	O
of	O
complex	O
analysis	O
in	O
providing	O
a	O
rich	O
family	O
of	O
holomorphic	O
functions	O
for	O
scientific	O
work	O
has	O
engaged	O
some	O
workers	O
in	O
efforts	O
to	O
extend	O
the	O
planar	O
theory	O
,	O
based	O
on	O
complex	O
numbers	O
,	O
to	O
a	O
4-space	O
study	O
with	O
functions	O
of	O
a	O
quaternion	B-Algorithm
variable	I-Algorithm
.	O
</s>
<s>
Then	O
is	O
an	O
extension	O
of	O
to	O
a	O
quaternion	B-Algorithm
variable	I-Algorithm
where	O
and	O
.	O
</s>
<s>
Since	O
the	O
time	O
of	O
Hamilton	O
,	O
it	O
has	O
been	O
realized	O
that	O
requiring	O
the	O
independence	O
of	O
the	O
derivative	B-Algorithm
from	O
the	O
path	O
that	O
a	O
differential	O
follows	O
toward	O
zero	O
is	O
too	O
restrictive	O
:	O
it	O
excludes	O
even	O
from	O
differentiation	O
.	O
</s>
<s>
Therefore	O
,	O
a	O
direction-dependent	O
derivative	B-Algorithm
is	O
necessary	O
for	O
functions	O
of	O
a	O
quaternion	B-Algorithm
variable	I-Algorithm
.	O
</s>
<s>
Considering	O
the	O
increment	O
of	O
polynomial	O
function	O
of	O
quaternionic	O
argument	O
shows	O
that	O
the	O
increment	O
is	O
a	O
linear	B-Architecture
map	I-Architecture
of	O
increment	O
of	O
the	O
argument	O
.	O
</s>
<s>
is	O
called	O
the	O
derivative	B-Algorithm
of	O
the	O
map	O
.	O
</s>
<s>
components	O
of	O
derivative	B-Algorithm
.	O
</s>
