--- title: "Laplace transform" sort_title: "Laplace transform" date: 2021-07-02 categories: - Mathematics - Physics layout: "concept" --- The **Laplace transform** is an integral transform that losslessly converts a function $$f(t)$$ of a real variable $$t$$, into a function $$\tilde{f}(s)$$ of a complex variable $$s$$, where $$s$$ is sometimes called the **complex frequency**, analogously to the [Fourier transform](/know/concept/fourier-transform/). The transform is defined as follows: $$\begin{aligned} \boxed{ \tilde{f}(s) \equiv \hat{\mathcal{L}}\{f(t)\} \equiv \int_0^\infty f(t) \exp(- s t) \dd{t} } \end{aligned}$$ Depending on $$f(t)$$, this integral may diverge. This is solved by restricting the domain of $$\tilde{f}(s)$$ to $$s$$ where $$\mathrm{Re}\{s\} > s_0$$, for an $$s_0$$ large enough to compensate for the growth of $$f(t)$$. The **inverse Laplace transform** $$\hat{\mathcal{L}}{}^{-1}$$ involves complex integration, and is therefore a lot more difficult to calculate. Fortunately, it is usually avoidable by rewriting a given $$s$$-space expression using [partial fraction decomposition](/know/concept/partial-fraction-decomposition/), and then looking up the individual terms. ## Derivatives The derivative of a transformed function is the transform of the original mutliplied by its variable. This is especially useful for transforming ODEs with variable coefficients: $$\begin{aligned} \boxed{ \tilde{f}{}'(s) = - \hat{\mathcal{L}}\{t f(t)\} } \end{aligned}$$ This property generalizes nicely to higher-order derivatives of $$s$$, so: $$\begin{aligned} \boxed{ \dvn{n}{\tilde{f}}{s} = (-1)^n \hat{\mathcal{L}}\{t^n f(t)\} } \end{aligned}$$