TABLE OF CONTENTS Many useful techniques depend on the Laplace transform. The Laplace transform of a function f( t) is denoted sometimes by L{f( t)} and sometimes by F( s). The inverse Laplace transform of F( s) is denoted sometimes by L -1{ F( s)} and sometimes by f( t). Figure 4.1 makes the relation clear; s is a complex variable whose role is defined by eqn. 4.1.
By definition
| (4.1) |  |
Let f( t) = a constant k, and let R( s) denote the real part of the complex number s
provided that R( s) is positive (for otherwise the integral does not exist).
Let f( t) = exp( at)
This will be true provided that R( s) > a.
The chore of calculating Laplace transforms of particular time functions and the converse problem - calculating the time function, by inverse Laplace transformation, corresponding with a particular Laplace transform - can be avoided by the use of software packages or tables of transform pairs. Small tables are to be found as appendices in many introductory control textbooks. A larger set of tables can be found in McCollum and Brown (1965) and a very comprehensive set in Prudnikov et al. (1992).
