# The Laplace Transformation

**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.

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## 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".

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Chapter Contents

- Laplace Transforms
- 1a. The Unit Step Function - Definition
- 1a.i. Oliver Heaviside
- 1b. The Unit Step Function - Products
- 2. Laplace Transform Definition
- 2a. Table of Laplace Transformations
- 3. Properties of Laplace Transform
- 4. Transform of Unit Step Functions
- 5. Transform of Periodic Functions
- 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