Introduction to Laplace Transforms and s-Domain Analysis
The Laplace transform is an integral transform that converts a function of a real variable t (often representing time) into a function of a complex variable s (representing complex frequency). Named after the mathematician Pierre-Simon Laplace, this transform is a cornerstone of engineering, physics, and applied mathematics. Its primary power lies in its ability to simplify differential equations — which govern dynamic physical systems — into simple algebraic equations, which are much easier to solve.
In time-domain analysis, solving differential equations involving derivatives, initial values, and driving forces can be mathematically tedious. By applying the Laplace transform, differentiation in the time domain is replaced by multiplication by s in the frequency domain. This allows engineers to solve for the system's transfer function, analyze stability, and then apply the inverse Laplace transform to return the solution to the time domain.
This calculator computes both forward Laplace transforms for standard functions and inverse Laplace transforms for proper rational s-domain functions, displaying the step-by-step partial fraction decompositions used during inversion.