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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 recommendations supplied.

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Solution Using the definition of Laplace transform, Now, we have that this last expression tends to Hence we have the result We can use this result to generalise as follows: Corollary Proof The proof is straightforward: If we put in this recurrence relation we obtain If we assume then This establishes that by induction. 2 Find the Laplace transform of and deduce the value of where is a real constant and a positive integer. Solution Using the first shift theorem with gives so with we get Using the formula follows.

4The Dirac- function We now ask ourselves what is the Laplace transform of ? Does it exist? 4) with , a perfectly valid choice of gives However, we progress with care. This is good advice when dealing with generalised functions. Let us take the Laplace transform of the top hat function defined mathematically by The calculation proceeds as follows:- As hence which as . In Laplace transform theory it is usual to define the impulse function such that This means reducing the width of the top hat function so that it lies between and (not and ) and increasing the height from to in order to preserve unit area.

All polynomial functions and (of course) exponential functions of the type ( constant) are included as well as bounded functions. Excluded functions are those that have singularities such as or and functions that have a growth rate more rapid than exponential, for example . Functions that have a finite number of finite discontinuities are also included. 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.

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