The Laplace transform is an integral transform that losslessly converts a function of a real variable , into a function of a complex variable , where is sometimes called the complex frequency, analogously to the Fourier transform. The transform is defined as follows:
Depending on , this integral may diverge. This is solved by restricting the domain of to where , for an large enough to compensate for the growth of .
The inverse Laplace transform involves complex integration, and is therefore a lot more difficult to calculate. Fortunately, it is usually avoidable by rewriting a given -space expression using partial fraction decomposition, and then looking up the individual terms.
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:
This property generalizes nicely to higher-order derivatives of , so:
The exponential is the only thing that depends on here:
The Laplace transform of a derivative introduces the initial conditions into the result. Notice that is the initial value in the original -domain:
This property generalizes to higher-order derivatives, although it gets messy quickly. Once again, the initial values of the lower derivatives appear:
Where is shorthand for the th derivative of , and . As an example, becomes .
We integrate by parts and use the fact that :
And so on. By partially integrating times in total we arrive at the conclusion.
- O. Bang, Applied mathematics for physicists: lecture notes, 2019, unpublished.