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 $s\in{\bf C}$ 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

$\displaystyle X(s) = \frac{A}{\alpha - s}, \quad ($ re $\displaystyle \left\{s\right\}>\alpha) \protect$

(D.1)

the Taylor series expansion of

about the point

in the

plane is given by

$\begin{eqnarray*} X(s) &=& X(s_0) + (s-s_0) X^\prime (s_0) + (s-s_0)^2\frac{X^... ...\ &\isdef & \sum_{n=0}^\infty (s-s_0)^n\frac{X^{(n)}(s_0)}{n!} \end{eqnarray*}$

where, writing as $(\alpha-s)^{-1}$ and using the chain rule for differentiation,

$\begin{eqnarray*} X^\prime (s_0) &\isdef & X^{(1)}(s_0) \isdef \left.\frac{d X(s... ...pha-s)^{-4}(-1)\right\vert _{s=s_0} = \frac{3!}{(\alpha-s)^4}\\ \end{eqnarray*}$

and so on. We also used the factorial notation $n!\isdeftext n(n-1)(n-2)\cdots 3\cdot 2\cdot 1$ , and we defined the special cases $0!\isdeftext 1$ and $X^{(0)}(s_0)\isdeftext X(s_0)$ , as is normally done. The series expansion of can thus be written

$\displaystyle X(s)$	$\displaystyle =$	$\displaystyle \frac{1}{\alpha-s_0} + \frac{s-s_0}{(\alpha-s_0)^2} + \frac{(s-s_0)^2}{(\alpha-s_0)^3} + \cdots$
	$\displaystyle =$	$\displaystyle \sum_{n=0}^\infty \frac{(s-s_0)^n}{(\alpha-s_0)^{n+1}}. \protect$	(D.2)

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

$\displaystyle X(\alpha) = \sum_{n=0}^\infty \frac{(\alpha-s_0)^n}{(\alpha-s_0)^{n+1}} = \sum_{n=0}^\infty \frac{1}{\alpha-s_0} = \infty \frac{1}{\alpha-s_0}.$

Thus, the series clearly does not converge for $s=\alpha$ , no matter what our choice of

might be. We must therefore accept the point at infinity for $H(\alpha)$ . This is eminently reasonable since the closed form Laplace transform we derived, $H(s) = 1/(\alpha - s)$ does ``blow up'' at $s=\alpha$ . The point $s=\alpha$ is called a pole of $H(s) = 1/(\alpha - s)$ .

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

$\displaystyle \frac{(s-s_0)^n}{(\alpha-s_0)^{n+1}}$

the ratio test demands that the ratio of term

over term

have absolute value less than

. That is, we require

$\displaystyle 1 > \left\vert \left. \frac{(s-s_0)^{n+1}}{(\alpha-s_0)^{n+2}} \r... ...lpha-s_0)^{n+1}}} \right\vert = \left\vert\frac{s-s_0}{\alpha-s_0}\right\vert,$

or,

$\displaystyle \zbox {\left\vert s-s_0\right\vert < \left\vert\alpha-s_0\right\vert.}$

We see that the region of convergence is a circle about the point

having radius approaching but not equal to $\vert\alpha-s_0\vert$ . Thus, the circular disk of convergence is centered at

and extends to, but does not touch, the pole at $s=\alpha$ .

The analytic continuation of the domain of Eq. $\,$ (D.1) is now defined as the union of the disks of convergence for all points $s_0\neq \alpha$ . 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 $s=\alpha$ .

In summary, the Laplace transform of an exponential $x(t)=A e^{\alpha t}$ is

$\displaystyle X(s) = \frac{A}{s-\alpha}$

and the value is well defined and finite for all $s\neq \alpha$ .

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.

Previous: Existence of the Laplace Transform
Next: Relation to the z Transform

Order a Hardcopy of Introduction to Digital Filters

About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

Sign in

Search Online Books

Free Online Books

Chapters

See Also

Introduction to Digital Filters
Introduction to Laplace Transform Analysis
Analytic Continuation