Download PDF by P.P.G. Dyke: An Introduction to Laplace Transforms and Fourier Series

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 recommendations supplied.

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Another appropriate quantity to find at this point is the determination of the value of the Laplace transform of First of all, the function must be integrable in such a way that is of exponential order. From this definition of it is immediately apparent that and that . This latter result is called the fundamental theorem of the calculus. We can now use the result to obtain where we have written . Hence which finally gives the result The following result is also useful and can be stated in the form of a theorem.

Of course it is possible to solve some kinds of differential equation without using convolution as is obvious from the last chapter, but mastery of the convolution theorem greatly extends the power of Laplace transforms to solve ODEs. 3. 2. Of course it is possible to solve some kinds of differential equation without using convolution as is obvious from the last chapter, but mastery of the convolution theorem greatly extends the power of Laplace transforms to solve ODEs. In fact, familiarity with the convolution operation is necessary for the understanding of many other topics such as the solution of partial differential equations (PDEs) and those that are outside the scope of this book such as the use of Green’s functions for forming the general solution of various types of boundary value problem (BVP).

We shall investigate this further later; meanwhile let us look at the Laplace transform of functions that are not even continuous. Functions that are not continuous occur naturally in branches of electrical and control engineering, and in the software industry. One only has to think of switches to realise how widespread discontinuous functions are throughout electronics and computing. 3 Find the Laplace transform of the function represented by where Solution This function is of the “saw-tooth” variety that is quite common in electrical engineering.

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