By P.P.G. Dyke

This complex undergraduate/graduate textbook offers an easy-to-read account of Fourier sequence, wavelets and Laplace transforms. It positive aspects many labored examples with all strategies supplied.

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Extra resources for An Introduction to Laplace Transforms and Fourier Series

Example text

1) gives Implement the change of variable in the inner integral so that it becomes Thus we have Hence This establishes the theorem. This particular result is sometimes referred to as Borel’s Theorem, and the convolution referred to as Faltung. These names are found in older books and some present day engineering texts. Before going on to use this theorem, let us do an example or two on calculating convolutions to get a feel of how the operation works. 1 Find the value of . Solution Using the definition of convolution we have To evaluate this we could of course resort to computer algebra: alternatively we use the identity This identity is engraved on the brains of those who passed exams before the advent of formula sheets.

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.

Solution Using the definition of convolution we have To evaluate this we could of course resort to computer algebra: alternatively we use the identity This identity is engraved on the brains of those who passed exams before the advent of formula sheets. Evaluating the integral by hand thus progresses as follows. Let and in this trigonometric formula to obtain whence, Let us try another example. 2 Find the value of . 2 . Of course we choose the order that gives the easier integral to evaluate. In fact there is little to choose in this present example, but it is a point worth watching in future.