Laplace transform and its inverse
The Laplace transform
The two-sided definition of the Laplace transform was encountered in . This is rarely used in engineering analysis, which prefers the following one-sided transform.1
Definition
Let \(f: \mathbb{R}_+ \rightarrow \mathbb{R}\) be a function of time \(t\) for which \(f(t)=0\) for \(t<0\). The Laplace transform \(\mathcal{L}: T\rightarrow S\) of \(f\) is defined as \[\begin{aligned} (\mathcal{L}f)(s) = \int_0^\infty f(t) e^{-s t}\ \diff t. \end{aligned}\]
As with the Fourier transform image, it is customary to capitalize the Laplace transform image; e.g.2 \[\begin{aligned} F(s) = (\mathcal{L}f)(s). \end{aligned}\]
As with the two-sided Laplace transform, if the transform exists, it will do so for some region of convergence (ROC), a subset of the \(s\)-plane. It is best practice to report a Laplace transform image paired with its ROC.
On the imaginary axis (\(\sigma = 0\)), \(s = j\omega\) and the Laplace transform is \[\begin{aligned} (\mathcal{L}f)(s) = \int_0^\infty f(t) e^{-j\omega t}\ \diff t, \end{aligned}\] which is the one-sided Fourier transform! Therefore, when the Laplace transform exists for a region of convergence that includes the imaginary axis, the one-sided Fourier transform also exists and3 \[\begin{aligned} (\mathcal{F}f)(\omega) = (\mathcal{L}f)(s)|_{s\mapsto j\omega} \end{aligned}\] or, haphazardly using \(F\) to denote both transforms, \[\begin{aligned} F(\omega) = F(s)|_{s\mapsto j\omega}. \end{aligned}\]
Laplace terminology
The terminology in the literature for the Laplace transform and its inverse, introduced next, is inconsistent. The “Laplace transform” is at once taken to be a function that maps a function of \(t\) to a function of \(s\) and a particular result of that mapping (technically the image of the map), which is a complex function of \(s\). Even we will say things like “the value of the Laplace transform \(F(s)\) at \(s = 2 + j 4\),” by which we really mean the image of the complex function \(F(s)|_{s \rightarrow 2 + j 4}\) that was the image of the Laplace transform map of the real function of time \(f(t)\). You can see why we shorten it.
The inverse Laplace transform
As with the Fourier transform, the Laplace transform has an inverse.
Definition
Let \(s \in \mathbb{C}\) be the Laplace \(s\) and \(F(s)\) a Laplace transform image of real function \(f(t)\). The inverse Laplace transform \(\mathcal{L}^{-1}: S \rightarrow T\) is defined as \[\begin{aligned} (\mathcal{L}^{-1}F)(t) = \frac{1} {j 2\pi} \int_{\sigma - j\infty}^{\sigma + j\infty} F(s) e^{s t}\ \diff s. \end{aligned}\]
As is illustrated in figure 10.1, it can be shown that the inverse Laplace transform image of a Laplace transform image of \(f(t)\) equals \(f(t)\) and vice-versa; i.e. \[\begin{aligned} (\mathcal{L}^{-1}\mathcal{L} f)(t) &= f(t) \quad\text{and}\\ (\mathcal{L}\mathcal{L}^{-1} F)(s) &= F(s). \end{aligned}\] That is, the inverse Laplace transform is a true inverse. Therefore, we call the Laplace transform and its inverse a pair.
A detail view of figure 10.1 is given in figure 10.2.
Returning to the troublesome unit step f(t) = us(t), calculate its Laplace transform image F(s).
Directly applying the definition, $$\begin{align} F(s) &= \int_0^\infty f(t) e^{-s t}\ \diff t \\ &= \cloze{\int_0^\infty 1 e^{-s t}\ \diff t} \\ &= \left.\frac{-1} {s} e^{-s t} \right|_{t=0}^\infty \\ &= \left( \lim_{t\rightarrow\infty} -e^{-(\sigma + j\omega) t}/s \right) - \frac{-1} {s} e^0 \tag{$s=\sigma+j\omega$} \\ &= 1/s. \tag{$\sigma>0$} \end{align}$$ Note that the limit only converges for σ > 0, so the region of convergence is the right half s-plane, exclusive of the imaginary axis. This exclusion tells us what we already know, that the Fourier transform us does not exist. However, the Laplace transform does exist and is simply 1/s!
Both the Laplace transform and especially its inverse are typically calculated with the help of software and tables such as , which includes specific images and important properties. We will first consider these properties in , then turn to the use of software and tables in where we focus on the more challenging inverse calculation.
Online Resources for Section 10.2
No online resources.