Categories: Mathematics, Physics.

Laplace transform

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. 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 $$f(t)$$.

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{ \dv[n]{\tilde{f}}{s} = (-1)^n \hat{\mathcal{L}}\{t^n f(t)\} } \end{aligned}

The Laplace transform of a derivative introduces the initial conditions into the result. Notice that $$f(0)$$ is the initial value in the original $$t$$-domain:

\begin{aligned} \boxed{ \hat{\mathcal{L}}\{ f'(t) \} = - f(0) + s \tilde{f}(s) } \end{aligned}

This property generalizes to higher-order derivatives, although it gets messy quickly. Once again, the initial values of the lower derivatives appear:

\begin{aligned} \boxed{ \hat{\mathcal{L}} \big\{ f^{(n)}(t) \big\} = - \sum_{j = 0}^{n - 1} s^j f^{(n - 1 - j)}(0) + s^n \tilde{f}(s) } \end{aligned}

Where $$f^{(n)}(t)$$ is shorthand for the $$n$$th derivative of $$f(t)$$, and $$f^{(0)}(t) = f(t)$$. As an example, $$\hat{\mathcal{L}}\{f'''(t)\}$$ becomes $$- f''(0) - s f'(0) - s^2 f(0) + s^3 \tilde{f}(s)$$.

1. O. Bang, Applied mathematics for physicists: lecture notes, 2019, unpublished.

© Marcus R.A. Newman, a.k.a. "Prefetch". Available under CC BY-SA 4.0.