Pierre-Simon Laplace
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The Laplace transform converts a function of a real variable (usually $t$ in the time domain) into a function of a complex variable $s$ (the complex frequency, often called the s-domain). Time-domain signals are typically denoted $f(t)$, while their Laplace transforms are written as $F(s)$ or $ℒ\left[f(t)\right]$.

The transform is useful because it converts differentiation and integration in the time domain into multiplication and division in the Laplace domain. Hence, ordinary differential equations and more complex integro-differential equations become algebraic equations that are easier to solve.

In electric circuits, Laplace methods are particularly important because the currents and voltages in inductors and capacitors are described by derivatives and integrals. Consequently, Kirchhoff's voltage law and Kirchhoff's current law lead to systems of integro-differential equations in the time domain that are relatively easy to solve 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.