## Analytic Continuation

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.

In the case of our exponential example

the Taylor series expansion of about the point in the plane is given by

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

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

*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

*pole*at .

The *analytic continuation* of the domain of Eq.(D.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

Analytic continuation works for any finite number of poles of finite
order,^{D.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.

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Relation to the z Transform

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Existence of the Laplace Transform