The Laplace transform transforms a function of a real variable, (usually $t$ in the time domain) to a function of a complex variable $s$ (in the complex-valued frequency domain, also known as s-domain). The functions are often denoted by $f(t)$ in the time domain representation, and $F(s)$ or $ℒ\left[f(t)\right]$ in the frequency-domain.

The transform is useful because it converts differentiation and integration in the time domain into much easier multiplication and division in the Laplace domain. Hence, ordinary differential equations or even the more complex <a href='javascript:void(0)' onclick='return myAlert(`Integro-differential equations`,`An integro-differential equation is an equation that involves both derivatives and integrals of an unknown function $f(t)$.`)'>integro-differential equation</a>s are converted to algebraic equations that are comparatively simple to solve.

In electric circuits, the method of Laplace is particularly important because the currents and voltages in inductors and capacitors are described by derivatives and integrals; therefore, <a href='javascript:void(0)' onclick='return kvl()'>Kirchhoff's voltage law</a> and <a href='javascript:void(0)' onclick='return kcl()'>Kirchhoff's current law</a> lead to systems of integro-differential equations in the time domain, which turn out to be solved relatively easy using Laplace transforms.

In this chapter we study the basic properties of the Laplace transform and learn how to compute the Laplace transform of simple functions of time, derivatives, and integrals. In the next chapter, we apply the rules that we learn here to study linear electric circuits containing inductors and capacitors in time-domain.