TABLE OF CONTENTS A Laplace transform is a mapping between the time domain and the domain of complex variable s defined by
where s is a complex variable, s= ?+ j ?, and f(t) is a sectionally continuous function of time. Function f(t) is also assumed to be equal to zero for t<0. With this assumption, the transform defined by Eq. (A2.1) is called a one-sided Laplace transform.
The condition for the existence of a Laplace transform of f(t) is that the integral in Eq. (A2.1) exists, which in turn requires that there exist real numbers, A and b, such that f(t)< Ae bt. Most functions of time encountered in engineering systems are Laplace transformable.
Laplace transforms are commonly used in solving linear differential equations. By application of the Laplace transform, the differential equations involving variables of time t are transformed into algebraic equations in the domain of complex variable s. The solutions of the algebraic equations, which are usually much easier to obtain than the solutions of the original differential equations, are then transformed back to the time domain by use of the inverse Laplace transform.
The inverse Laplace transform is defined by the Riemann integral:
The Riemann integral is rarely used in practice. The most common practical method used in inverse Laplace transformation is the method of partial fraction expansion, which will be described later.
A short...
