Don’t let anyone tell you all this is easy! There is a great deal of sophisticated mathematics
to deal with in this problem, and I have spelt it out carefully now because you will need
to do a lot of it later – I wanted to get the main difficulties out of the way up front so that
we are not too distracted later on. We will now get down to the main business of defining
the Laplace transform.
In a similar way to how we transform avariableto simplify some mathematical problem
such as integration, we can often similarly transform afunctionto facilitate the solution
to a differential equation. The most common way to do this is by means of an integral.
Thus, iff(t)is a function oft(it is usual to usetas the independent variable, since in
applications it usually refers to time), then we transform to a new functionf(s) ̃ of a new
variablesby an expression of the form:
f(s) ̃ =∫bak(s,t)f(t)dtThe notation ‘ftwiddle’,f(s) ̃ , is a standard mathematical way of reminding us that the
new function ofscomes fromf, but is completely different tof, which is a function oft.
In Problem 17.1 we hadf(t)=t,k(s,t)=e−st,a=0andb=∞, and the integral, which
would correspond to what we have calledf(s) ̃ here, came out to be
1
s^2. The above integral
expression is called anintegral transform. Essentially, it replaces the functionf(t)by a
different functionf(s) ̃ , which may be easier to deal with in certain circumstances.
There are many such integral transforms, usually going under the name of some famous
mathematician – Laplace, Fourier, Mellin, Hilbert. The Laplace and Fourier transforms
are particularly useful, in scientific and engineering applications such as control theory
and signal analysis, and in statistical and commercial applications. I would certainly not
waste your time by introducing such complicated objects if they were not so useful! It is
worth noting that such transforms were often first invented and developed by scientists
and engineers, long before mathematicians got their hands on them.
When we consider a transform such as the Laplace transform we are interested in such
questions as:
- What are the Laplace transforms of the elementary functions?
- What general properties does the transform have?
- How does the Laplace transform relate to other mathematical operations
(such as differentiation, integration, etc.)? - How can the transform be ‘inverted’, i.e. given the transformf(s) ̃ of
a functionf(t), how do we reverse the transform and find the original
functionf(t)? - What applications does the Laplace transform have?
We now give the formal definition of the Laplace transform. We have already looked at
the technical details of integration needed to deal with it, in Problem 17.1, so it should be
less of a shock! Be assured that, strange though it might appear, it is absolutely invaluable
in engineering mathematics.
Supposef(t)is a function defined fort≥0. Then the integral
f(s) ̃ =∫∞0f(t)e−stdt