<s>
In	O
mathematics	O
,	O
in	O
the	O
area	O
of	O
complex	O
analysis	O
,	O
Nachbin	B-Algorithm
's	I-Algorithm
theorem	I-Algorithm
(	O
named	O
after	O
Leopoldo	O
Nachbin	O
)	O
is	O
commonly	O
used	O
to	O
establish	O
a	O
bound	O
on	O
the	O
growth	O
rates	O
for	O
an	O
analytic	B-Language
function	I-Language
.	O
</s>
<s>
This	O
article	O
provides	O
a	O
brief	O
review	O
of	O
growth	O
rates	O
,	O
including	O
the	O
idea	O
of	O
a	O
function	B-Algorithm
of	I-Algorithm
exponential	I-Algorithm
type	I-Algorithm
.	O
</s>
<s>
Classification	O
of	O
growth	O
rates	O
based	O
on	O
type	O
help	O
provide	O
a	O
finer	O
tool	O
than	O
big	O
O	O
or	O
Landau	O
notation	O
,	O
since	O
a	O
number	O
of	O
theorems	O
about	O
the	O
analytic	O
structure	O
of	O
the	O
bounded	O
function	O
and	O
its	O
integral	B-Algorithm
transforms	I-Algorithm
can	O
be	O
stated	O
.	O
</s>
<s>
In	O
particular	O
,	O
Nachbin	B-Algorithm
's	I-Algorithm
theorem	I-Algorithm
may	O
be	O
used	O
to	O
give	O
the	O
domain	O
of	O
convergence	O
of	O
the	O
generalized	B-Algorithm
Borel	I-Algorithm
transform	I-Algorithm
,	O
given	O
below	O
.	O
</s>
<s>
Nachbin	B-Algorithm
's	I-Algorithm
theorem	I-Algorithm
has	O
immediate	O
applications	O
in	O
Cauchy	O
theorem-like	O
situations	O
,	O
and	O
for	O
integral	B-Algorithm
transforms	I-Algorithm
.	O
</s>
<s>
The	O
integral	O
form	O
for	O
the	O
generalized	B-Algorithm
Borel	I-Algorithm
transform	I-Algorithm
follows	O
as	O
well	O
.	O
</s>
<s>
Nachbin	B-Algorithm
resummation	I-Algorithm
(	O
generalized	B-Algorithm
Borel	I-Algorithm
transform	I-Algorithm
)	O
can	O
be	O
used	O
to	O
sum	O
divergent	B-Algorithm
series	I-Algorithm
that	O
escape	O
to	O
the	O
usual	O
Borel	O
summation	O
or	O
even	O
to	O
solve	O
(	O
asymptotically	O
)	O
integral	O
equations	O
of	O
the	O
form	O
:	O
</s>
<s>
where	O
f(t )	O
may	O
or	O
may	O
not	O
be	O
of	O
exponential	O
growth	O
and	O
the	O
kernel	B-Algorithm
K(u )	O
has	O
a	O
Mellin	O
transform	O
.	O
</s>
