# Chapter one

Mathematical skills

## Section five: Laplace Transforms

This chapter is on the theme of supporting mathematics that are needed for engineering problem solving, analysis, design and evaluation.

This section focuses on Laplace transforms. We approach this as an engineer who needs to use the tool rather than giving a rigorous development which you would get in a mathematics course.

In practice, we recommend that once you have sufficient understanding, use computer tools to do any numerical computations.

### 1. Simple signals

The most important and simplest signals are a step and an exponential.

A talk through video is on YouTube. Read the notes (PDF, 535KB).

### 2. Sinusoidal signals and the shift theorem

Introducing Laplace for sinusoidal signals and the shift theorem, thus also considering sinusoids scaled by an exponential.

A talk through video is on YouTube. *Note the typing error at 9 minutes 40 seconds, where '4' is written instead of '42'.* Read the notes (PDF, 547KB).

### 3. Handling derivatives

This shows how the definition of Laplace leads to the transform for an nth order derivative. It shows how one can apply Laplace to determine the solution for differential equations.

A talk through video is on YouTube. Read the notes (PDF, 555KB).

### 4. Summary

The summary gives a table of Laplace transforms of common signals. It also gives a cursory summary of some topics that may be useful occasionally, such as the Laplace of an integral, a delayed signal and impulse function. There are some self test tutorial questions.

A talk through video is on YouTube. Read the notes (PDF, 605KB).

### 5. Tutorial sheet

Introductory tutorial sheet with outline solutions.

View the tutorial sheet (PDF, 566KB).