Denoted as , the Laplace transform transforms a function from time domain to complex frequency domain. A common notation for the Laplace transform is to represent the transformed function as the uppercase version of the original time-domain function:
Where is the original function in the time domain, and is the transformed function in the complex frequency domain, with being a complex variable.
not to be confused with the Lie derivative, which is also commonly denoted
Note that the result of a Laplace transform includes not only the algebraic expression but also the region of convergence (ROC).
The unilateral Laplace transform is the most commonly used form of the Laplace transform and is typically what people refer to as “the” Laplace transform.
Below are some of the properties of the unilateral Laplace transform. The properties of bilateral Laplace transform are similar, but there are some important differences.
Insight: we can view as , and indeed they get the consistent result.
Derive
Note that
Laplace Transform to Solve Differential Equation
The Laplace transform is a powerful tool for solving differential equations. When applied to a differential equation, it typically converts the problem into an algebraic equation that we can readily solve with algebra.