The Laplace transform takes a function of time and transforms it to a function of a complex variable \(s\). Because the transform is invertible, no information is lost and it is reasonable to think of a function \(f(t)\) and its Laplace transform \(F(s)\) as two views of the same phenomenon. Each view has its uses and some features of the phenomenon are easier to understand in one view or the other.
We can use the Laplace transform to transform a linear time invariant system from the time domain to the \(s\)-domain. This leads to the system function \(G(s)\) for the system –this is the same system function used in the Nyquist criterion for stability.
One important feature of the Laplace transform is that it can transform analytic problems to algebraic problems. We will see examples of this for differential equations.
- 13.3: Exponential Type
- The Laplace transform is defined when the integral for it converges. Functions of exponential type are a class of functions for which the integral converges for all s with Re(s) large enough.
- 13.7: System Functions and the Laplace Transform
- When we introduced the Nyquist criterion for stability we stated without any justification that the system was stable if all the poles of the system function G(s) were in the left half-plane. We also asserted that the poles corresponded to exponential modes of the system. In this section we’ll use the Laplace transform to more fully develop these ideas for differential equations.
- 13.8: Laplace inverse
- Up to now we have computed the inverse Laplace transform by table lookup. To do this properly we should first check that the Laplace transform has an inverse. We start with the bad news: Unfortunately this is not strictly true. There are many functions with the same Laplace transform.