<s>
In	O
mathematics	O
,	O
superfunction	B-Algorithm
is	O
a	O
nonstandard	O
name	O
for	O
an	O
iterated	B-Algorithm
function	I-Algorithm
for	O
complexified	O
continuous	O
iteration	O
index	O
.	O
</s>
<s>
Then	O
,	O
S( z	O
;	O
x	O
)	O
can	O
be	O
interpreted	O
as	O
the	O
superfunction	B-Algorithm
of	O
the	O
function	O
f(x )	O
.	O
</s>
<s>
Much	O
study	O
and	O
many	O
applications	O
of	O
superfunctions	B-Algorithm
employ	O
various	O
extensions	O
of	O
these	O
superfunctions	B-Algorithm
to	O
complex	O
and	O
continuous	O
indices	O
;	O
and	O
the	O
analysis	O
of	O
existence	O
,	O
uniqueness	O
and	O
their	O
evaluation	O
.	O
</s>
<s>
The	O
Ackermann	O
functions	O
and	O
tetration	O
can	O
be	O
interpreted	O
in	O
terms	O
of	O
superfunctions	B-Algorithm
.	O
</s>
<s>
Analysis	O
of	O
superfunctions	B-Algorithm
arose	O
from	O
applications	O
of	O
the	O
evaluation	O
of	O
fractional	B-Algorithm
iterations	I-Algorithm
of	O
functions	O
.	O
</s>
<s>
Superfunctions	B-Algorithm
and	O
their	O
inverses	O
allow	O
evaluation	O
of	O
not	O
only	O
the	O
first	O
negative	O
power	O
of	O
a	O
function	O
(	O
inverse	O
function	O
)	O
,	O
but	O
also	O
of	O
any	O
real	O
and	O
even	O
complex	O
iterate	O
of	O
that	O
function	O
.	O
</s>
<s>
and	O
extend	O
the	O
range	O
of	O
definition	O
of	O
the	O
superfunction	B-Algorithm
S	O
to	O
the	O
non-negative	O
integers	O
.	O
</s>
<s>
In	O
particular	O
,	O
tetration	O
can	O
be	O
interpreted	O
as	O
superfunction	B-Algorithm
of	O
exponentiation	O
for	O
some	O
real	O
base	O
;	O
in	O
this	O
case	O
,	O
</s>
<s>
For	O
extension	O
to	O
non-integer	O
values	O
of	O
the	O
argument	O
,	O
the	O
superfunction	B-Algorithm
should	O
be	O
defined	O
in	O
a	O
different	O
way	O
.	O
</s>
<s>
In	O
general	O
,	O
the	O
superfunction	B-Algorithm
is	O
not	O
unique	O
.	O
</s>
<s>
The	O
modified	O
superfunction	B-Algorithm
may	O
have	O
a	O
narrower	O
range	O
of	O
holomorphy	O
.	O
</s>
<s>
The	O
variety	O
of	O
possible	O
superfunctions	B-Algorithm
is	O
especially	O
large	O
in	O
the	O
limiting	O
case	O
,	O
when	O
the	O
width	O
of	O
the	O
range	O
of	O
holomorphy	O
becomes	O
zero	O
;	O
in	O
this	O
case	O
,	O
one	O
deals	O
with	O
real-analytic	O
superfunctions	B-Algorithm
.	O
</s>
<s>
If	O
the	O
range	O
of	O
holomorphy	O
required	O
is	O
large	O
enough	O
,	O
then	O
the	O
superfunction	B-Algorithm
is	O
expected	O
to	O
be	O
unique	O
,	O
</s>
<s>
This	O
short	O
collection	O
of	O
elementary	O
superfunctions	B-Algorithm
is	O
illustrated	O
in	O
.	O
</s>
<s>
Some	O
superfunctions	B-Algorithm
can	O
be	O
expressed	O
through	O
elementary	O
functions	O
;	O
</s>
<s>
they	O
are	O
used	O
without	O
mention	O
that	O
they	O
are	O
superfunctions	B-Algorithm
.	O
</s>
<s>
the	O
superfunction	B-Algorithm
is	O
just	O
addition	O
of	O
a	O
constant	O
.	O
</s>
<s>
Exponentiation	O
is	O
superfunction	B-Algorithm
(	O
from	O
1	O
to	O
)	O
of	O
function	O
.	O
</s>
<s>
is	O
a	O
superfunction	B-Algorithm
(	O
iteration	B-Algorithm
orbit	I-Algorithm
)	O
of	O
f	O
.	O
</s>
<s>
and	O
the	O
superfunction	B-Algorithm
approaches	O
unity	O
in	O
the	O
negative	O
direction	O
on	O
the	O
real	O
axis	O
:	O
</s>
<s>
The	O
tetration	O
is	O
then	O
a	O
superfunction	B-Algorithm
of	O
.	O
</s>
<s>
The	O
inverse	O
of	O
a	O
superfunction	B-Algorithm
for	O
a	O
suitable	O
argument	O
x	O
can	O
be	O
interpreted	O
as	O
the	O
Abel	O
function	O
,	O
the	O
solution	O
of	O
the	O
Abel	O
equation	O
,	O
</s>
<s>
upgrade	O
of	O
the	O
floating	B-Algorithm
point	I-Algorithm
representation	I-Algorithm
of	O
numbers	O
in	O
computers	O
.	O
</s>
<s>
Any	O
holomorphic	O
function	O
can	O
be	O
identified	O
to	O
a	O
transfer	O
function	O
,	O
and	O
then	O
its	O
superfunctions	B-Algorithm
and	O
corresponding	O
Abel	O
functions	O
can	O
be	O
considered	O
.	O
</s>
<s>
The	O
reconstruction	O
of	O
the	O
superfunction	B-Algorithm
from	O
the	O
transfer	O
function	O
allows	O
to	O
work	O
with	O
relatively	O
thick	O
samples	O
,	O
improving	O
the	O
precision	O
of	O
measurements	O
.	O
</s>
<s>
the	O
transfer	O
function	O
,	O
may	O
be	O
boosted	O
with	O
the	O
superfunction	B-Algorithm
.	O
</s>
<s>
The	O
mass	O
of	O
the	O
snowball	O
could	O
be	O
measured	O
at	O
the	O
top	O
of	O
the	O
hill	O
and	O
at	O
the	O
bottom	O
,	O
giving	O
the	O
transfer	O
function	O
;	O
then	O
,	O
the	O
mass	O
of	O
the	O
snowball	O
,	O
as	O
a	O
function	O
of	O
the	O
length	O
it	O
passed	O
,	O
is	O
a	O
superfunction	B-Algorithm
.	O
</s>
<s>
Such	O
a	O
function	O
can	O
be	O
evaluated	O
through	O
the	O
superfunction	B-Algorithm
and	O
the	O
Abel	O
function	O
of	O
the	O
transfer	O
function	O
.	O
</s>
