By P.P.G. Dyke
This complex undergraduate/graduate textbook presents an easy-to-read account of Fourier sequence, wavelets and Laplace transforms. It positive factors many labored examples with all suggestions supplied.
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Extra info for An Introduction to Laplace Transforms and Fourier Series
1. Trivial rewriting of this double integral to facilitate changing the order of integration gives Fig. 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.
Sometimes it is necessary to define what is called the two sided Laplace transform which makes a great deal of mathematical sense. However the additional problems that arise by allowing negative values of are severe and limit the use of the two sided Laplace transform. For this reason, the two sided transform will not be pursued here. Fig. 1 , a function with no well defined starting value Fig. 2 Derivative Property of the Laplace Transform Suppose a differentiable function has Laplace transform, we can find the Laplace transform of its derivative through the following theorem.
1 If an interval say can be partitioned into a finite number of subintervals with an increasing sequence of times and such that a given function is continuous in each of these subintervals but not necessarily at the end points themselves, then is piecewise continuous in the interval . Only functions that differ at a finite number of points have the same Laplace transform. If except at a finite number of points where they differ by finite values then . We mention this again in the next chapter when the inverse Laplace transform is defined.
An Introduction to Laplace Transforms and Fourier Series by P.P.G. Dyke