# Laplace Transform

## Table of Contents

## 1. Introduction

The dual-edge Laplace Transform is defined as:

\begin{align} \label{Laplace Transform} F(s) = \int_{-\infty}^{\infty}f(t)e^{-st}dt \end{align}when \(s\) is complex (which it usually is), the Fourier Transform can be taken to be a special case of the dual-edge Laplace Transform. One can think of this as analyzing the complex exponential domain, rather than just the frequency domain (imaginary exponential domain). Now, multiplying the signal by the Heaviside Step Function:

\begin{align} \label{Step Function} F(s) = \int_{-\infty}^{\infty}H(t)f(t)e^{-st}dt = \int_{0}^{\infty}f(t)e^{-st}dt \end{align}gives you the conventional Laplace Transform.

### 1.1. Usage

The Laplace Transform is primarily used for analyzing linear differential equations as it converts these equations into algebraic equations. The inverse Laplace Transform is complicated, and is therefore not used often. Instead, Laplace Transform tables are used in order to convert back into the time-domain. Taking the Laplace transform of all terms in a linear differential equation will yield this result. One of the simplest differential equations that the Laplace Transform can solve is the mass-spring system, and it also generally has applications in circuit analysis.