Our frequency-domain analysis has been limited to circuits with sinusoidal inputs. In other words, we have assumed sinusoidal time-varying excitations in all our non-dc circuits so The Laplace transform is signiﬁcant for a number of reasons. First, it can be applied to a wider variety of inputs than phasor analysis. Second, it provides an easy way to solve circuit problems involving initial conditions, because it allows us to work with algebraic equations instead of differential equations. Third, the Laplace transform is capable of providing us, in one single operation, the total response of the circuit comprising both the natural and forced responses
Given a function f (t), its Laplace transform, denoted by F(s) or L [f (t)], is given by
Where s is a complex variable given by s = σ +jω.