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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 factors many labored examples with all strategies supplied.

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Example text

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. 1 (Linearity) If and are two functions whose Laplace transform exists, then where and are arbitrary constants. Proof where we have assumed that so that where This proves the theorem. Here we shall concentrate on those properties of the Laplace transform that do not involve the calculus. The first of these takes the form of another theorem because of its generality.

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Hence which finally gives the result The following result is also useful and can be stated in the form of a theorem. 3 If then assuming that Proof Let be the function , so that . Using the property we deduce that Integrating both sides of this with respect to from to gives since which completes the proof. The function defines the Sine Integral function which occurs in the study of optics. The formula for its Laplace transform can now be easily derived as follows. 3 Heaviside’s Unit Step Function As promised earlier, we devote this section to exploring some properties of Heaviside’s unit step function .

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