Understanding Engineering Mathematics

• continuous and discontinuous functions (418
  ➤
  )


• partial fractions (60
  ➤
  )


• completing the square (60
  ➤
  )


• differential equations (Chapter 15)


• sigma notation (102

➤

)

• infinite series (428

➤

)

Objectives

In this chapter you will find:

• definition of the Laplace transform


• Laplace transforms of elementary functions


• properties of Laplace transforms


• solution of initial value problems by Laplace transform


• the inverse Laplace transform


• linear systems and superposition


• definition of Fourier series


• orthogonality of trig functions


• determination of the coefficients in a Fourier series

Motivation

You may need the material of this chapter for:

• solving differential equations in electrical circuits


• analysing the behaviour of control systems


• signal analysis

17.2 The Laplace transform

Problem 17.1

Integrate the following, where s and a are constants

∫a

0

te−stdt

Leta→∞in the result, assuming thats>0.

This integral has a number of features that we need to look at to get us into the mood
for Laplace transforms. Expressions that contain a number of symbols, some constant and
some variable, often appear daunting to the most experienced of us. Just take it steady
and pick the bones out of the thing. Firstly, sinceaandsare to be regarded as constant