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# 13: Laplace Transform

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The Laplace transform takes a function of time and transforms it to a function of a complex variable $$s$$. Because the transform is invertible, no information is lost and it is reasonable to think of a function $$f(t)$$ and its Laplace transform $$F(s)$$ as two views of the same phenomenon. Each view has its uses and some features of the phenomenon are easier to understand in one view or the other.

We can use the Laplace transform to transform a linear time invariant system from the time domain to the $$s$$-domain. This leads to the system function $$G(s)$$ for the system –this is the same system function used in the Nyquist criterion for stability.

One important feature of the Laplace transform is that it can transform analytic problems to algebraic problems. We will see examples of this for differential equations.

13: Laplace Transform is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Jeremy Orloff via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.