<s>
In	O
the	O
theory	B-Architecture
of	I-Architecture
functions	I-Architecture
of	I-Architecture
several	I-Architecture
complex	I-Architecture
variables	I-Architecture
,	O
Hartogs	B-Algorithm
's	I-Algorithm
extension	I-Algorithm
theorem	I-Algorithm
is	O
a	O
statement	O
about	O
the	O
singularities	O
of	O
holomorphic	B-Architecture
functions	I-Architecture
of	I-Architecture
several	I-Architecture
variables	I-Architecture
.	O
</s>
<s>
Informally	O
,	O
it	O
states	O
that	O
the	O
support	O
of	O
the	O
singularities	O
of	O
such	O
functions	O
cannot	O
be	O
compact	O
,	O
therefore	O
the	O
singular	O
set	O
of	O
a	O
function	B-Architecture
of	I-Architecture
several	I-Architecture
complex	I-Architecture
variables	I-Architecture
must	O
(	O
loosely	O
speaking	O
)	O
'	O
go	O
off	O
to	O
infinity	O
 '	O
in	O
some	O
direction	O
.	O
</s>
<s>
More	O
precisely	O
,	O
it	O
shows	O
that	O
an	O
isolated	O
singularity	O
is	O
always	O
a	O
removable	O
singularity	O
for	O
any	O
analytic	B-Language
function	I-Language
of	O
complex	O
variables	O
.	O
</s>
<s>
A	O
first	O
version	O
of	O
this	O
theorem	O
was	O
proved	O
by	O
Friedrich	O
Hartogs	O
,	O
and	O
as	O
such	O
it	O
is	O
known	O
also	O
as	O
Hartogs	B-Algorithm
's	I-Algorithm
lemma	I-Algorithm
and	O
Hartogs	O
's	O
principle	O
:	O
in	O
earlier	O
Soviet	O
literature	O
,	O
it	O
is	O
also	O
called	O
Osgood	B-Algorithm
–	I-Algorithm
Brown	I-Algorithm
theorem	I-Algorithm
,	O
acknowledging	O
later	O
work	O
by	O
Arthur	O
Barton	O
Brown	O
and	O
William	O
Fogg	O
Osgood	O
.	O
</s>
<s>
This	O
property	O
of	O
holomorphic	B-Architecture
functions	I-Architecture
of	I-Architecture
several	I-Architecture
variables	I-Architecture
is	O
also	O
called	O
Hartogs	O
's	O
phenomenon	O
:	O
however	O
,	O
the	O
locution	O
"	O
Hartogs	O
's	O
phenomenon	O
"	O
is	O
also	O
used	O
to	O
identify	O
the	O
property	O
of	O
solutions	O
of	O
systems	O
of	O
partial	O
differential	O
or	O
convolution	B-Algorithm
equations	O
satisfying	O
Hartogs	O
type	O
theorems	O
.	O
</s>
<s>
The	O
original	O
proof	O
was	O
given	O
by	O
Friedrich	O
Hartogs	O
in	O
1906	O
,	O
using	O
Cauchy	O
's	O
integral	O
formula	O
for	O
functions	B-Architecture
of	I-Architecture
several	I-Architecture
complex	I-Architecture
variables	I-Architecture
.	O
</s>
<s>
Today	O
,	O
usual	O
proofs	O
rely	O
on	O
either	O
the	O
Bochner	B-Algorithm
–	I-Algorithm
Martinelli	I-Algorithm
–	I-Algorithm
Koppelman	I-Algorithm
formula	I-Algorithm
or	O
the	O
solution	O
of	O
the	O
inhomogeneous	O
Cauchy	O
–	O
Riemann	O
equations	O
with	O
compact	O
support	O
.	O
</s>
<s>
Yet	O
another	O
very	O
simple	O
proof	O
of	O
this	O
result	O
was	O
given	O
by	O
Gaetano	O
Fichera	O
in	O
the	O
paper	O
,	O
by	O
using	O
his	O
solution	O
of	O
the	O
Dirichlet	O
problem	O
for	O
holomorphic	B-Architecture
functions	I-Architecture
of	I-Architecture
several	I-Architecture
variables	I-Architecture
and	O
the	O
related	O
concept	O
of	O
CR-function	O
:	O
later	O
he	O
extended	O
the	O
theorem	O
to	O
a	O
certain	O
class	O
of	O
partial	O
differential	O
operators	O
in	O
the	O
paper	O
,	O
and	O
his	O
ideas	O
were	O
later	O
further	O
explored	O
by	O
Giuliano	O
Bratti	O
.	O
</s>
<s>
Such	O
a	O
phenomenon	O
is	O
called	O
Hartogs	O
's	O
phenomenon	O
,	O
which	O
lead	O
to	O
the	O
notion	O
of	O
this	O
Hartogs	B-Algorithm
's	I-Algorithm
extension	I-Algorithm
theorem	I-Algorithm
and	O
the	O
domain	B-Algorithm
of	I-Algorithm
holomorphy	I-Algorithm
.	O
</s>
<s>
Therefore	O
,	O
the	O
Hartogs	O
's	O
phenomenon	O
is	O
an	O
elementary	O
phenomenon	O
that	O
highlights	O
the	O
difference	O
between	O
the	O
theory	O
of	O
functions	O
of	O
one	O
and	O
several	B-Architecture
complex	I-Architecture
variables	I-Architecture
.	O
</s>
