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The Laplace transformation is a powerful method for the solution of linear differential equations. Although Laplace had used integrals involving exponential functions for this purpose at the beginning of the 19th century, the method described here was developed about 100 years later in connection with Heaviside's operational calculus.
The aim of this work is to show how the Laplace transformation can be applied to a variety of problems. Laplace transforms have many useful properties, and I have attempted to give valid mathematical proofs of these. Many of the properties are quite easy to establish, but some are much harder to prove rigorously. Readers interested only in the applications may wish to pass over the more difficult proofs others may prefer to leave them for a later reading.
Passages which may well be omitted or postponed are marked by a line in the margin these passages are intended primarily for reference purposes. Scientists and engineers should be aware that mathematical tools have limitations: mathematicians (who are not necessarily distinct persons from these scientists and engineers) should know why these limitations are there in order to give advice in cases of difficulty