### Definition of Laplace Transform

Let $f(t)$ be a given function which is defined for $t \ge 0$. If there exists a function $F(s)$ so that

then $F(s)$ is called the **Laplace Transform** of $f(t)$, and will be denoted by $\mathcal{L} \left\{f(t)\right\}$. Notice the integrator $e^{-st} \, dt$ where $s$ is a parameter which may be real or complex.

Thus,

The symbol $\mathcal{L}$ which transform $f(t)$ into $F(s)$ is called the Laplace transform operator.

Laplace transformation is a powerful method of solving linear differential equations. It reduces the problem of solving differential equations into algebraic equations. For more information about the application of Laplace transform in engineering, see this Wikipedia article and this Wolfram article.

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