Laplace transformation

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The Laplace Transform: Theory and Applications

Joel L. Schiff


To my parents


It is customary to begin courses in mathematical engineering by explaining that the lecturer would never trust his life to an aeroplane whose behaviour depended on properties of the Lebesgue integral. It might, perhaps, be just as foolhardy to fly in an aeroplane designed by an engineer whobelieved that cookbook application of the Laplace transform revealed all that was to be known about its stability. T.W. K¨ rner o Fourier Analysis Cambridge University Press 1988



The Laplace transform is a wonderful tool for solving ordinary and partial differential equations and has enjoyed much success in this realm. With its success, however, a certain casualness has beenbred concerning its application, without much regard for hypotheses and when they are valid. Even proofs of theorems often lack rigor, and dubious mathematical practices are not uncommon in the literature for students. In the present text, I have tried to bring to the subject a certain amount of mathematical correctness and make it accessible to undergraduates. To this end, this text addresses anumber of issues that are rarely considered. For instance, when we apply the Laplace transform method to a linear ordinary differential equation with constant coefficients, an y(n) + an−1 y(n−1) + · · · + a0 y f (t),

why is it justified to take the Laplace transform of both sides of the equation (Theorem A.6)? Or, in many proofs it is required to take the limit inside an integral. This is alwaysfrought with danger, especially with an improper integral, and not always justified. I have given complete details (sometimes in the Appendix) whenever this procedure is required.




Furthermore, it is sometimes desirable to take the Laplace transform of an infinite series term by term. Again it is shown that this cannot always be done, and specific sufficient conditions areestablished to justify this operation. Another delicate problem in the literature has been the application of the Laplace transform to the so-called Dirac delta function. Except for texts on the theory of distributions, traditional treatments are usually heuristic in nature. In the present text we give a new and mathematically rigorous account of the Dirac delta function based upon theRiemann–Stieltjes integral. It is elementary in scope and entirely suited to this level of exposition. One of the highlights of the Laplace transform theory is the complex inversion formula, examined in Chapter 4. It is the most sophisticated tool in the Laplace transform arsenal. In order to facilitate understanding of the inversion formula and its many subsequent applications, a self-contained summary of thetheory of complex variables is given in Chapter 3. On the whole, while setting out the theory as explicitly and carefully as possible, the wide range of practical applications for which the Laplace transform is so ideally suited also receive their due coverage. Thus I hope that the text will appeal to students of mathematics and engineering alike. Historical Summary. Integral transforms date back tothe work of L´ onard Euler (1763 and 1769), who considered them essentially in e the form of the inverse Laplace transform in solving second-order, linear ordinary differential equations. Even Laplace, in his great work, Th´orie analytique des probabilit´s (1812), credits Euler with e e introducing integral transforms. It is Spitzer (1878) who attached the name of Laplace to the expression


esx φ(s) ds

employed by Euler. In this form it is substituted into the differential equation where y is the unknown function of the variable x. In the late 19th century, the Laplace transform was extended to its complex form by Poincar´ and Pincherle, rediscovered by Petzval, e



and extended to two variables by Picard, with further investigations conducted by Abel...
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