17 An Appreciation of Transform Methods
17.1 Introduction
We have already met situations in which changing the variable in a mathematical problem
works wonders – as for example in substitution in integration. In fact, this idea oftrans-
formingvariables to a new set is a useful one throughout applied mathematics. In this
chapter we are going to look at two particular types of transforms that are virtually essential
in engineering and science:
- Laplace transform
- Fourier series (transform)
Both serve useful roles in two key engineering topics:
- the study of initial value problems in control theory, where one is inter-
ested in stability properties of a system - harmonic analysis of signals in which a periodic input to a system
is decomposed into sinusoidal components which may be individually
processed and the results recombined to produce the output of the system
Although I will try to explain how the transforms arise, the topic may still require some-
thing of a leap of faith on your part, particularly in the definitions of the Laplace transform
and the Fourier series. This is a place where accepting the results blindly and understanding
later is not necessarily a bad learning strategy! Also, we will not prove very much in detail.
The priority will be to equip you with the main tools and give you an appreciation of the
concepts and methods. This topic is also useful in that it brings together so much of
the basic mathematical material that we have covered in this book, perhaps justifying
all the hard work put in!
Prerequisites
It will be helpful if you know something about- the exponential function (Chapter 4)
- integration, particularly integration by parts (Chapter 9)
- trig functions (Chapter 6)
- integration of products of sines and cosines (270
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) - limits at infinity (417
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) - function notation (90
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)