The Laplace transform of a function $f(t)$ is defined as $$\begin{equation}ℒ\left[f(t)\right]=F(\class{mjblue}{s})=\int_{0}^{\infty}f(t)e^{-\class{mjblue}{s} t} dt\end{equation}$$ where $\class{mjblue}{s}$ is the complex frequency and function $f(t)$ is assumed to be defined for $t\geq 0$. The Laplace transform is linear and homogeneous $$\begin{equation}ℒ\left[a f(t)+b g(t)\right]=a F(\class{mjblue}{s}) + b G(\class{mjblue}{s})\end{equation}$$ (where $a$ and $b$ are constants and $F$ and $G$ are the Laplace transforms of $f$ and $g$). The transform has the uniqueness property that, for a given suitable $f(t)$, there is a unique $F(\class{mjblue}{s})$.

The Laplace transform can be used to solve integro-differential equations that appear in electric circuit analysis. It reduces a linear differential, integral, or integro-differential equation to an algebraic equation, which can then be solved with standard algebraic methods. Finally, the solution of the original equation is obtained by applying the inverse Laplace transform to the algebraic solution.