<s>
In	O
mathematics	O
,	O
the	O
inverse	B-Algorithm
Laplace	I-Algorithm
transform	I-Algorithm
of	O
a	O
function	O
F(s )	O
is	O
the	O
piecewise-continuous	O
and	O
exponentially-restricted	O
real	O
function	O
f(t )	O
which	O
has	O
the	O
property	O
:	O
</s>
<s>
It	O
can	O
be	O
proven	O
that	O
,	O
if	O
a	O
function	O
F(s )	O
has	O
the	O
inverse	B-Algorithm
Laplace	I-Algorithm
transform	I-Algorithm
f(t )	O
,	O
then	O
f(t )	O
is	O
uniquely	O
determined	O
(	O
considering	O
functions	O
which	O
differ	O
from	O
each	O
other	O
only	O
on	O
a	O
point	O
set	O
having	O
Lebesgue	O
measure	O
zero	O
as	O
the	O
same	O
)	O
.	O
</s>
<s>
This	O
result	O
was	O
first	O
proven	O
by	O
Mathias	O
Lerch	O
in	O
1903	O
and	O
is	O
known	O
as	O
Lerch	B-Algorithm
's	I-Algorithm
theorem	I-Algorithm
.	O
</s>
<s>
The	O
Laplace	O
transform	O
and	O
the	O
inverse	B-Algorithm
Laplace	I-Algorithm
transform	I-Algorithm
together	O
have	O
a	O
number	O
of	O
properties	O
that	O
make	O
them	O
useful	O
for	O
analysing	O
linear	O
dynamical	O
systems	O
.	O
</s>
<s>
An	O
integral	O
formula	O
for	O
the	O
inverse	B-Algorithm
Laplace	I-Algorithm
transform	I-Algorithm
,	O
called	O
the	O
Mellin	O
's	O
inverse	O
formula	O
,	O
the	O
Bromwich	O
integral	O
,	O
or	O
the	O
Fourier	O
–	O
Mellin	O
integral	O
,	O
is	O
given	O
by	O
the	O
line	O
integral	O
:	O
</s>
<s>
Post	O
's	O
inversion	O
formula	O
for	O
Laplace	O
transforms	O
,	O
named	O
after	O
Emil	O
Post	O
,	O
is	O
a	O
simple-looking	O
but	O
usually	O
impractical	O
formula	O
for	O
evaluating	O
an	O
inverse	B-Algorithm
Laplace	I-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
With	O
the	O
advent	O
of	O
powerful	O
personal	O
computers	O
,	O
the	O
main	O
efforts	O
to	O
use	O
this	O
formula	O
have	O
come	O
from	O
dealing	O
with	O
approximations	O
or	O
asymptotic	O
analysis	O
of	O
the	O
Inverse	B-Algorithm
Laplace	I-Algorithm
transform	I-Algorithm
,	O
using	O
the	O
Grunwald	O
–	O
Letnikov	O
differintegral	O
to	O
evaluate	O
the	O
derivatives	O
.	O
</s>
