# Convolution Formula: Proof, Connection with Laplace Transform, Application to Physical Problems

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Lesson Summary:

In this lecture, the concept of convolution is introduced as the process of using two functions to get a third function. The formula for convolution is given as the integral of one function multiplied by the other function shifted by the dummy variable, and the Laplace transform of the two initial functions is shown to be related to the resulting convolution. Examples are provided to illustrate the use of convolution, and the commutative property of convolution is derived. Additionally, the proof of convolution is demonstrated using double integrals and the definitions of Laplace transforms.

Lesson Description:

Convolution Formula: Proof, Connection with Laplace Transform, Application to Physical Problems -- Lecture 21. Some more discussion, proof, and application for the Laplace Transform.

Arthur Mattuck, 18.03 Differential Equations, Spring 2006. (Massachusetts Institute of Technology: MIT OpenCourseWare), http://ocw.mit.edu (Accessed November 27, 2008). License: Creative Commons BY-NC-SA.

• What is the convolution formula?
• What is the Laplace Transform?
• How do you use the Laplace Transform?
• What are some applications of the Laplace Transform?
• What is f * g?
• Why does f * g = g * f?
• Are Laplace Transforms commutative?
• What is the inverse Laplace Transform?
• What are some applications of the convolution function?
• #### Staff Review

• Currently 4.0/5 Stars.
A really difficult and involved lecture. Many applications of the Laplace Transform are discussed, and the convolution formula is explained. Properties of the transform are also discussed. You will have a much deeper understanding of the Laplace Transform after this lecture.