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.