<s>
Zero	B-Algorithm
to	I-Algorithm
the	I-Algorithm
power	I-Algorithm
of	I-Algorithm
zero	I-Algorithm
,	O
denoted	O
by	O
,	O
is	O
a	O
mathematical	O
expression	O
that	O
is	O
either	O
defined	O
as	O
1	O
or	O
left	O
undefined	O
,	O
depending	O
on	O
context	O
.	O
</s>
<s>
The	O
combinatorial	O
interpretation	O
of	O
is	O
the	O
number	O
of	O
0-tuples	B-Application
of	O
elements	O
from	O
a	O
-element	O
set	O
;	O
there	O
is	O
exactly	O
one	O
0-tuple	O
.	O
</s>
<s>
Limits	O
involving	O
algebraic	O
operations	O
can	O
often	O
be	O
evaluated	O
by	O
replacing	O
subexpressions	O
by	O
their	O
limits	O
;	O
if	O
the	O
resulting	O
expression	O
does	O
not	O
determine	O
the	O
original	O
limit	O
,	O
the	O
expression	O
is	O
known	O
as	O
an	O
indeterminate	B-Algorithm
form	I-Algorithm
.	O
</s>
<s>
The	O
expression	O
is	O
an	O
indeterminate	B-Algorithm
form	I-Algorithm
:	O
Given	O
real-valued	O
functions	O
and	O
approaching	O
(	O
as	O
approaches	O
a	O
real	O
number	O
or	O
)	O
with	O
,	O
the	O
limit	O
of	O
can	O
be	O
any	O
non-negative	O
real	O
number	O
or	O
,	O
or	O
it	O
can	O
diverge	B-Algorithm
,	O
depending	O
on	O
and	O
.	O
</s>
<s>
For	O
example	O
,	O
each	O
limit	O
below	O
involves	O
a	O
function	O
with	O
as	O
(	O
a	O
one-sided	B-Algorithm
limit	I-Algorithm
)	O
,	O
but	O
their	O
values	O
are	O
different	O
:	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
if	O
and	O
are	O
analytic	B-Language
functions	I-Language
on	O
an	O
open	O
neighborhood	O
of	O
a	O
number	O
,	O
then	O
as	O
approaches	O
from	O
any	O
side	O
on	O
which	O
is	O
positive	O
.	O
</s>
<s>
In	O
1814	O
,	O
Pfaff	O
used	O
a	O
squeeze	B-Algorithm
theorem	I-Algorithm
argument	O
to	O
prove	O
that	O
as	O
.	O
</s>
<s>
With	O
this	O
justification	O
,	O
he	O
listed	O
along	O
with	O
expressions	O
like	O
in	O
a	O
table	O
of	O
indeterminate	B-Algorithm
forms	I-Algorithm
.	O
</s>
<s>
Möbius	O
reduced	O
to	O
the	O
case	O
,	O
but	O
then	O
made	O
the	O
mistake	O
of	O
assuming	O
that	O
each	O
of	O
and	O
could	O
be	O
expressed	O
in	O
the	O
form	O
for	O
some	O
continuous	O
function	O
not	O
vanishing	O
at	O
and	O
some	O
nonnegative	O
integer	O
,	O
which	O
is	O
true	O
for	O
analytic	B-Language
functions	I-Language
,	O
but	O
not	O
in	O
general	O
.	O
</s>
<s>
Knuth	O
(	O
1992	O
)	O
contends	O
more	O
strongly	O
that	O
"	O
has	O
to	O
be	O
"	O
;	O
he	O
draws	O
a	O
distinction	O
between	O
the	O
value	O
,	O
which	O
should	O
equal	O
,	O
and	O
the	O
limiting	O
form	O
(	O
an	O
abbreviation	O
for	O
a	O
limit	O
of	O
where	O
)	O
,	O
which	O
is	O
an	O
indeterminate	B-Algorithm
form	I-Algorithm
:	O
"	O
Both	O
Cauchy	O
and	O
Libri	O
were	O
right	O
,	O
but	O
Libri	O
and	O
his	O
defenders	O
did	O
not	O
understand	O
why	O
truth	O
was	O
on	O
their	O
side.	O
"	O
</s>
<s>
Other	O
authors	O
leave	O
undefined	O
because	O
is	O
an	O
indeterminate	B-Algorithm
form	I-Algorithm
:	O
does	O
not	O
imply	O
.	O
</s>
<s>
pow	O
(	O
whose	O
intent	O
is	O
to	O
return	O
a	O
non-NaN	O
result	O
when	O
the	O
exponent	O
is	O
an	O
integer	O
,	O
like	O
pown	O
)	O
treats	O
as	O
.	O
</s>
<s>
powr	O
treats	O
as	O
NaN	O
(	O
Not-a-Number	O
)	O
due	O
to	O
the	O
indeterminate	B-Algorithm
form	I-Algorithm
;	O
see	O
.	O
</s>
<s>
The	O
pow	O
variant	O
is	O
inspired	O
by	O
the	O
pow	O
function	O
from	O
C99	B-Language
,	O
mainly	O
for	O
compatibility	O
.	O
</s>
<s>
But	O
for	O
C	O
,	O
as	O
of	O
C99	B-Language
,	O
if	O
the	O
normative	O
annexF	O
is	O
supported	O
,	O
the	O
result	O
for	O
real	O
floating-point	O
types	O
is	O
required	O
to	O
be	O
because	O
there	O
are	O
significant	O
applications	O
for	O
which	O
this	O
value	O
is	O
more	O
useful	O
than	O
NaN	O
(	O
for	O
instance	O
,	O
with	O
discrete	O
exponents	O
)	O
;	O
the	O
result	O
on	O
complex	O
types	O
is	O
not	O
specified	O
,	O
even	O
if	O
the	O
informative	O
annexG	O
is	O
supported	O
.	O
</s>
<s>
The	O
Java	B-Language
standard	O
,	O
the	O
.NET	B-Application
Framework	I-Application
method	B-Language
System.Math.Pow	O
,	O
Julia	B-Application
,	O
and	O
Python	B-Language
also	O
treat	O
as	O
.	O
</s>
<s>
Some	O
languages	O
document	O
that	O
their	O
exponentiation	O
operation	O
corresponds	O
to	O
the	O
pow	O
function	O
from	O
the	O
C	B-Language
mathematical	I-Language
library	I-Language
;	O
this	O
is	O
the	O
case	O
with	O
Lua	B-Language
and	O
Perl	B-Language
's	O
**	O
operator	O
(	O
where	O
it	O
is	O
explicitly	O
mentioned	O
that	O
the	O
result	O
of	O
0**0	B-Algorithm
is	O
platform-dependent	O
)	O
.	O
</s>
<s>
APL	B-Language
,	O
R	B-Language
,	O
Stata	B-Algorithm
,	O
SageMath	B-Application
,	O
Matlab	B-Language
,	O
Magma	B-General_Concept
,	O
GAP	B-General_Concept
,	O
Singular	B-General_Concept
,	O
PARI/GP	B-Language
,	O
and	O
GNU	B-Language
Octave	I-Language
evaluate	O
to	O
.	O
</s>
<s>
Mathematica	B-Language
and	O
Macsyma	B-Language
simplify	O
to	O
even	O
if	O
no	O
constraints	O
are	O
placed	O
on	O
;	O
however	O
,	O
if	O
is	O
entered	O
directly	O
,	O
it	O
is	O
treated	O
as	O
an	O
error	O
or	O
indeterminate	O
.	O
</s>
<s>
SageMath	B-Application
does	O
not	O
simplify	O
.	O
</s>
<s>
Maple	B-Language
,	O
Mathematica	B-Language
and	O
PARI/GP	B-Language
further	O
distinguish	O
between	O
integer	O
and	O
floating-point	O
values	O
:	O
If	O
the	O
exponent	O
is	O
a	O
zero	O
of	O
integer	O
type	O
,	O
they	O
return	O
a	O
of	O
the	O
type	O
of	O
the	O
base	O
;	O
exponentiation	O
with	O
a	O
floating-point	O
exponent	O
of	O
value	O
zero	O
is	O
treated	O
as	O
undefined	O
,	O
indeterminate	O
or	O
error	O
.	O
</s>
