The Laplace transforms for various time functions will now be considered. These are readily obtainable through a direct application of Eq. (2.54).
For the unit step function defined by
the Laplace transform is
Therefore,
From here on we assume that f(t) = 0 for t < 0.
For the function
Therefore,
For the function
the Laplace transform is
Integrating by parts,
with u = t, dv = e−st dt, the following is obtained:
Therefore,
For the function
The Laplace transform is
The solution to Eq. (2.63) is simplified by using the exponential form of sin ωt,
Therefore,
Therefore,
Once the Laplace transform for any function f(t) is obtained and tabulated, it need not be derived again. The foregoing results and other important transform pairs useful to the control engineer appear in Table 2.1. An extended table is shown in Appendix A. In addition, the location of the poles of the transformed function in the s-plane is listed in Table 2.1.