<s>
In	O
the	O
mathematical	O
theory	O
of	O
functions	O
of	O
one	O
or	O
more	B-Architecture
complex	I-Architecture
variables	I-Architecture
,	O
and	O
also	O
in	O
complex	O
algebraic	O
geometry	O
,	O
a	O
biholomorphism	B-Algorithm
or	O
biholomorphic	B-Algorithm
function	I-Algorithm
is	O
a	O
bijective	B-Algorithm
holomorphic	O
function	O
whose	O
inverse	O
is	O
also	O
holomorphic	O
.	O
</s>
<s>
Formally	O
,	O
a	O
biholomorphic	B-Algorithm
function	I-Algorithm
is	O
a	O
function	O
defined	O
on	O
an	O
open	O
subset	O
U	O
of	O
the	O
-dimensional	O
complex	O
space	O
Cn	O
with	O
values	O
in	O
Cn	O
which	O
is	O
holomorphic	O
and	O
one-to-one	B-Algorithm
,	O
such	O
that	O
its	O
image	O
is	O
an	O
open	O
set	O
in	O
Cn	O
and	O
the	O
inverse	O
is	O
also	O
holomorphic	O
.	O
</s>
<s>
As	O
in	O
the	O
case	O
of	O
functions	O
of	O
a	O
single	O
complex	O
variable	O
,	O
a	O
sufficient	O
condition	O
for	O
a	O
holomorphic	O
map	O
to	O
be	O
biholomorphic	B-Algorithm
onto	O
its	O
image	O
is	O
that	O
the	O
map	O
is	O
injective	O
,	O
in	O
which	O
case	O
the	O
inverse	O
is	O
also	O
holomorphic	O
(	O
e.g.	O
,	O
see	O
Gunning	O
1990	O
,	O
Theorem	O
I.11	O
)	O
.	O
</s>
<s>
If	O
there	O
exists	O
a	O
biholomorphism	B-Algorithm
,	O
we	O
say	O
that	O
U	O
and	O
V	O
are	O
biholomorphically	B-Algorithm
equivalent	I-Algorithm
or	O
that	O
they	O
are	O
biholomorphic	B-Algorithm
.	O
</s>
<s>
If	O
every	O
simply	O
connected	O
open	O
set	O
other	O
than	O
the	O
whole	O
complex	O
plane	O
is	O
biholomorphic	B-Algorithm
to	O
the	O
unit	O
disc	O
(	O
this	O
is	O
the	O
Riemann	O
mapping	O
theorem	O
)	O
.	O
</s>
<s>
For	O
example	O
,	O
open	O
unit	O
balls	O
and	O
open	O
unit	O
polydiscs	B-Algorithm
are	O
not	O
biholomorphically	B-Algorithm
equivalent	I-Algorithm
for	O
In	O
fact	O
,	O
there	O
does	O
not	O
exist	O
even	O
a	O
proper	O
holomorphic	O
function	O
from	O
one	O
to	O
the	O
other	O
.	O
</s>
<s>
According	O
to	O
this	O
definition	O
,	O
a	O
map	O
f	O
:	O
U	O
→	O
C	O
is	O
conformal	O
if	O
and	O
only	O
if	O
f	O
:	O
U	O
→	O
f(U )	O
is	O
biholomorphic	B-Algorithm
.	O
</s>
<s>
Notice	O
that	O
per	O
definition	O
of	O
biholomorphisms	B-Algorithm
,	O
nothing	O
is	O
assumed	O
about	O
their	O
derivatives	O
,	O
so	O
,	O
this	O
equivalence	O
contains	O
the	O
claim	O
that	O
a	O
homeomorphism	O
that	O
is	O
complex	O
differentiable	O
must	O
actually	O
have	O
nonzero	O
derivative	O
everywhere	O
.	O
</s>
<s>
According	O
to	O
this	O
weaker	O
definition	O
,	O
a	O
conformal	O
map	O
need	O
not	O
be	O
biholomorphic	B-Algorithm
,	O
even	O
though	O
it	O
is	O
locally	O
biholomorphic	B-Algorithm
,	O
for	O
example	O
,	O
by	O
the	O
inverse	O
function	O
theorem	O
.	O
</s>
<s>
For	O
example	O
,	O
if	O
f	O
:	O
U	O
→	O
U	O
is	O
defined	O
by	O
f(z )	O
=	O
z2	O
with	O
U	O
=	O
C–{0},	O
then	O
f	O
is	O
conformal	O
on	O
U	O
,	O
since	O
its	O
derivative	O
f’( z	O
)	O
=	O
2z	O
≠	O
0	O
,	O
but	O
it	O
is	O
not	O
biholomorphic	B-Algorithm
,	O
since	O
it	O
is	O
2-1	O
.	O
</s>
