It turns out that the domain of definition of the Laplace transform can be extended
by means of analytic continuation [14, p. 259].
Analytic continuation is carried out by expanding a function of
about all points in its domain of definition, and
extending the domain of definition to all points for which the series
expansion converges.

where, writing as
and using the chain rule for
differentiation,

and so on. We also used the factorial notation
, and we defined the special cases
and
, as is normally done.
The series expansion of can thus be written

(B.2)

We now ask for what values of does the series Eq. (B.2)
converge? The value is particularly easy to
check, since

Thus, the series clearly does not converge for , no
matter what our choice of might be. We must therefore accept
the point at infinity for . This is eminently reasonable
since the closed form Laplace transform we derived,
does ``blow up'' at . The point is called a
pole of
.

More generally, let's apply the ratio test for the convergence
of a geometric series. Since the th term of the series is

the ratio test demands that the ratio of term over
term have absolute value less than . That is, we
require

or,

We see that the region of convergence is a circle about the point
having radius approaching but not equal to
. Thus, the circular disk of convergence is
centered at and extends to, but does not touch, the
pole at .

The analytic continuation of the domain of Eq. (B.1) is now
defined as the union of the disks of convergence for all points
. It is easy to see that a sequence of such disks can
be chosen so as to define all points in the plane except at the
pole .

In summary, the Laplace transform of an exponential
is

and the value is well defined and finite for all
.

Analytic continuation works for any finite number of poles of finite
order,^{B.2} and for an infinite number of
distinct poles of finite order. It breaks down only in pathological
situations such as when the Laplace transform is singular everywhere
on some closed contour in the complex plane. Such pathologies do not
arise in practice, so we need not be concerned about them.