<s>
In	O
mathematics	O
,	O
the	O
two-sided	B-Algorithm
Laplace	I-Algorithm
transform	I-Algorithm
or	O
bilateral	B-Algorithm
Laplace	I-Algorithm
transform	I-Algorithm
is	O
an	O
integral	B-Algorithm
transform	I-Algorithm
equivalent	O
to	O
probability	O
's	O
moment	O
generating	O
function	O
.	O
</s>
<s>
Two-sided	B-Algorithm
Laplace	I-Algorithm
transforms	I-Algorithm
are	O
closely	O
related	O
to	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
,	O
the	O
Mellin	O
transform	O
,	O
the	O
Z-transform	B-Algorithm
and	O
the	O
ordinary	O
or	O
one-sided	O
Laplace	O
transform	O
.	O
</s>
<s>
In	O
population	O
ecology	O
,	O
the	O
argument	O
t	O
often	O
represents	O
spatial	O
displacement	O
in	O
a	O
dispersal	O
kernel	B-Algorithm
.	O
</s>
<s>
The	O
Fourier	B-Algorithm
transform	I-Algorithm
can	O
be	O
defined	O
in	O
terms	O
of	O
the	O
two-sided	B-Algorithm
Laplace	I-Algorithm
transform	I-Algorithm
:	O
</s>
<s>
The	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
normally	O
defined	O
so	O
that	O
it	O
exists	O
for	O
real	O
values	O
;	O
the	O
above	O
definition	O
defines	O
the	O
image	O
in	O
a	O
strip	O
which	O
may	O
not	O
include	O
the	O
real	O
axis	O
where	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
is	O
supposed	O
to	O
converge	O
.	O
</s>
<s>
This	O
is	O
then	O
why	O
Laplace	O
transforms	O
retain	O
their	O
value	O
in	O
control	O
theory	O
and	O
signal	O
processing	O
:	O
the	O
convergence	O
of	O
a	O
Fourier	B-Algorithm
transform	I-Algorithm
integral	O
within	O
its	O
domain	O
only	O
means	O
that	O
a	O
linear	O
,	O
shift-invariant	O
system	O
described	O
by	O
it	O
is	O
stable	O
or	O
critical	O
.	O
</s>
<s>
At	O
the	O
same	O
time	O
,	O
nowadays	O
Laplace	O
transform	O
theory	O
falls	O
within	O
the	O
ambit	O
of	O
more	O
general	O
integral	B-Algorithm
transforms	I-Algorithm
,	O
or	O
even	O
general	O
harmonical	O
analysis	O
.	O
</s>
<s>
+	O
Properties	O
of	O
the	O
bilateral	B-Algorithm
Laplace	I-Algorithm
transform	I-Algorithm
Property	O
Time	O
domain	O
domain	O
Strip	O
of	O
convergence	O
Comment	O
Definition	O
Time	O
scaling	O
Reversal	O
Frequency-domain	O
derivative	B-Algorithm
Frequency-domain	O
general	O
derivative	B-Algorithm
Derivative	B-Algorithm
General	O
derivative	B-Algorithm
Frequency-domain	O
integration	O
only	O
valid	O
if	O
the	O
integral	O
exists	O
Time-domain	O
integral	O
Time-domain	O
integral	O
Frequency	O
shifting	O
Time	O
shifting	O
Modulation	O
Finite	O
difference	O
Multiplication	O
.	O
</s>
<s>
Most	O
properties	O
of	O
the	O
bilateral	B-Algorithm
Laplace	I-Algorithm
transform	I-Algorithm
are	O
very	O
similar	O
to	O
properties	O
of	O
the	O
unilateral	O
Laplace	O
transform	O
,	O
</s>
<s>
This	O
theorem	O
is	O
proved	O
by	O
applying	O
the	O
inverse	O
Laplace	O
transform	O
on	O
the	O
convolution	B-Language
theorem	O
in	O
form	O
of	O
the	O
cross-correlation	O
.	O
</s>
<s>
For	O
any	O
two	O
functions	O
for	O
which	O
the	O
two-sided	B-Algorithm
Laplace	I-Algorithm
transforms	I-Algorithm
exist	O
,	O
if	O
i.e.	O
</s>
<s>
The	O
Laplace	O
transform	O
is	O
analytic	B-Language
in	O
the	O
region	O
of	O
absolute	O
convergence	O
.	O
</s>
<s>
In	O
particular	O
,	O
it	O
is	O
analytic	B-Language
.	O
</s>
<s>
Bilateral	O
transforms	O
do	O
not	O
respect	O
causality	B-Application
.	O
</s>
