# Week 3 - Laplace Transform

Updated 2017-10-10

**Context**: when dealing with control systems, taking derivatives and integration is common, but is too error-prone, computation-intensive, and complicated. The Laplace transform of a function will allow us to deal with the system in **frequency domain**. As a result, derivation and integration turns into multiplication and division of \(s\), the complex frequency.

## Definition

The definition of full Laplace transform is given as:

\[\mathcal L:\int_{-\infty}^\infty f(t)e^{-st}\mathrm dt\]The function is integrated from time being \(-\infty\) to \(\infty\). However, we don’t care about what happens the time far far before. We care about the system after time 0. Thus we have the *half* Laplace transform:

## List of Laplace Transforms

**Integration**

**Derivation**

**Delta Function**

Where \(\delta(t)\) is a “infinite spike” at \(t=0\), and \(u(t)\) is the unit step function.