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Chapter 4:
The Laplace Transform
Although the Fourier transform has been shown to be a useful tool in a variety of applications, there are others for which it is not particularly well suited. Generally these other applications involve initial value problems for which the auxiliary data are prescribed at t=0 . Also, many of the functions which commonly arise in engineering and science applications -€” like sinusoidal functions and polynomials — do not have Fourier transforms in the usual sense without the introduction of generalized functions. For these reasons, among others, it is useful to develop other integral transforms. There are numerous integral transforms that have been developed over the years, many of which are highly specialized. The most versatile of all integral transforms, including the Fourier transform, is the Laplace transform. Laplace transforms date back to the French mathematician Laplace who made use of the transform integral in his work on probability theory in the 1780s. S. D. Poisson (1781-1840) also knew of the Laplace transform integral in the 1820s and it occurred in Fourier's famous 1811 paper on heat conduction.
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