We have had several occasions to use Laplace transforms in this text. This appendix examines useful properties of the Laplace transform and gives a table of Laplace transforms that are often used. For further details about Laplace transform, see Schiff [SCHI 1999] or Bellman and Roth [BELL 1994].

Suppose that is a piecewise-continuous function, defined at least for , and is of *exponential* order , meaning that it does not grow any faster than the exponential :

for some constant *M*. Then the Laplace transform of , denoted by , is defined by the integral

for any complex number s such that its real part Re.

Important pairs are given in Table D.1.

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