Applied Mathematics/Laplace Transforms

The Laplace transform is an integral transform which is widely used in physics and engineering. Laplace Transforms involve a technique to change an expression into another form that is easier to work with using an improper integral. We usually introduce Laplace Transforms in the context of differential equations, since we use them a lot to solve some differential equations that can't be solved using other standard techniques. However, Laplace Transforms require only improper integration techniques to use. So you may run across them in first year calculus. Notation: The Laplace Transform is denoted as . The Laplace transform is named after mathematician and astronomer Pierre-Simon Laplace. Topics You Need to Understand For This Page Improper Integrals Definition For a function , using Napier's constant and a complex number , the Laplace transform is defined as follows: The parameter is a complex number. with real numbers and . This is the Laplace transform of . Explanation Here is what is going on. Examples of Laplace transform + Examples of Laplace transform (n>0) In the above table, and are constants is a natural number is the Delta function is the Heaviside function Examples 1. Calculate (where is a constant) using the integral definition. 2. Calculate using the integral definition.