With respect to the range of convergence of the Laplace integral
If the function to be transformed 
is stepwise continuous
and if there are real numbers 
and
such
that for all 
 |
with
. Particularly, if for
the smallest possible value
is taken, the condition
is the smallest possible convergence range.
Consequently the Laplace integral exists only within some part of
the complex plane. This
part is called convergence area, as shown in Figure A.1. The
variable
is called
the abscissa of
convergence.
For values of with
Eq. (A.1) makes no sense.
Thus for
the limit value of
for
must go to zero, but not for
.
 |
with
, as
for
decreases faster than any
power of 
increases.
