By P.P.G. Dyke
This complex undergraduate/graduate textbook offers an easy-to-read account of Fourier sequence, wavelets and Laplace transforms. It good points many labored examples with all strategies supplied.
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Additional info for An Introduction to Laplace Transforms and Fourier Series
Numerical inversion techniques are possible and these can be found in some software packages, especially those used by control engineers. Insight into the behaviour of the solution can be deduced without actually solving the differential equation by examining the asymptotic character of for small or large . In fact, it is often very useful to determine this asymptotic behaviour without solving the equation, even when exact solutions are available as these solutions are often complex and difficult to obtain let alone interpret.
These have a special role in the theory of Laplace transforms so we will not dwell on them here: suffice to say that a function such as is one example. However, the function is excluded because although all the discontinuities are finite, there are infinitely many of them. We shall now follow standard practice and use (time) instead of as the dummy variable. 3 Elementary Properties The Laplace transform has many interesting and useful properties, the most fundamental of which is linearity. It is linearity that enables us to add results together to deduce other more complicated ones and is so basic that we state it as a theorem and prove it first.
The Laplace transform of is therefore Here is another useful general result; we state it as a theorem. 3 If and in general Proof Let us start with the definition of Laplace transform and differentiate this with respect to to give assuming absolute convergence to justify interchanging differentiation and (improper) integration. Hence One can now see how to progress by induction. Assume the result holds for , so that and differentiate both sides with respect to (assuming all appropriate convergence properties) to give or So which establishes the result by induction.
An Introduction to Laplace Transforms and Fourier Series by P.P.G. Dyke