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
, 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
More generally, let's apply the ratio test for the convergence of a geometric series. Since the th term of the series is
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.
Relation to the z Transform
Existence of the Laplace Transform