Covering all aspects of Laplace Transforms that could be covered in a first semester Differential Equations curriculum.
Many first semester Differential Equations class accross universities will not be able to move at a pace that can cover all the topics deemed by consensus as Introductory Differential Equations Material. Instead of currating a cirriculum that satisfies one classroom's requirements, the goal was to take a union of all Laplace Transform material taught in an applied introductory class and let the student parse the material they will need ad hoc.
This covers many material aspects needed for other heavy science classes like Convolution, Step Functions, Gamma and Beta Functions, Dirac Delta Functions. Each topic including Laplace Transforms themselves are taught at a ground up level to motivate each definition.
While we do bring in some high school level Complex Analysis shortcuts, most introduction Differential Equations Classes do not require Complex Analysis as a Pre-requisite. So there is no investigation into Inverse Laplace Transforms or proofs for say its Existence and Uniqueness theorem.