Modern Control System Theory and Design, 2nd Edition

2.4.  REVIEW OF THE LAPLACE TRANSFORM

The Laplace transform [1–5] is helpful in the solution of ordinary differential equations describing the behavior of systems. When the transform operates on a differential equation, a transformed equation results. It is expressed in terms of an arbitrary complex variable s. The resulting transformed equation is in purely algebraic terms, which can be easily manipulated to obtain a solution for the desired quantity as an explicit function of the complex variable. In order to obtain a solution in terms of the original variable, it is necessary to carry out an inversion process to determine the desired time function. The inverse Laplace transform is given by

where c is a real constant greater than the real part of any singularity of F(s). The evaluation of this integral is usually difficult, and the inverse transformations are usually obtained by expanding F(s) into simpler components using a partial fraction expansion, and then obtaining the equivalent time-domain representation of the simpler components utilizing a table of Laplace transforms. The Laplace transform F(s) of a certain function of time f(t) is conventionally written as

From the definition of the Laplace transform given by Eq. (2.54), the integral exists if