The Laplace Transformation

Pierre-Simon Laplace

Pierre-Simon Laplace (1749-1827)

Laplace was a French mathematician, astronomer, and physicist who applied the Newtonian theory of gravitation to the solar system (an important problem of his day). He played a leading role in the development of the metric system.

The Laplace Transform is widely used in engineering applications (mechanical and electronic), especially where the driving force is discontinuous. It is also used in process control.

What Does the Laplace Transform Do?

The main idea behind the Laplace Transformation is that we can solve an equation (or system of equations) containing differential and integral terms by transforming the equation in "t-space" to one in "s-space". This makes the problem much easier to solve. The kinds of problems where the Laplace Transform is invaluable occur in electronics. You can take a sneak preview in the Applications of Laplace section.

If needed we can find the inverse Laplace transform, which gives us the solution back in "t-space".

What's in this Chapter?

1a. The Unit Step Function - Definition (otherwise known as Heaviside function)

1b. The Unit Step Function - Products (how to "turn on" or "turn off" signals at different times)

2. Laplace Transform Definition (as an infinite integral)

Table of Laplace Transformations (an easier way to find Laplace Transforms)

3. Properties of Laplace Transform (with worked examples)

4. Transform of Unit Step Functions

5. Transform of Periodic Functions (like sine and cosine)

6. Transforms of Integrals

7. Inverse of the Laplace Transform

8. Using Inverse Laplace to Solve DEs

9. Integro-Differential Equations and Systems of DEs

10. Applications of Laplace Transform


We begin the chapter by introducing the topic of Unit Step Functions »

NOTE: This Chapter is entirely done using Scientific Notebook logo. Ordinary Web pages are also provided for your convenience.

Didn't find what you are looking for on this page? Try search:

Online Algebra Solver

This algebra solver can solve a wide range of math problems. (Please be patient while it loads.)

Ready for a break?


Play a math game.

(Well, not really a math game, but each game was made using math...)

The IntMath Newsletter

Sign up for the free IntMath Newsletter. Get math study tips, information, news and updates each fortnight. Join thousands of satisfied students, teachers and parents!

Given name: * required

Family name:

email: * required

See the Interactive Mathematics spam guarantee.

Share IntMath!

Short URL for this Page

Save typing! You can use this URL to reach this page:

Calculus Lessons on DVD


Easy to understand calculus lessons on DVD. See samples before you commit.

More info: Calculus videos