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## Convergence

With respect to the range of convergence of the Laplace integral

 (A.1)

now the following considerations are taken:

If the function to be transformed is stepwise continuous and if there are real numbers and such that for all

is valid, then the Laplace integral will converge for all with . Particularly, if for the smallest possible value is taken, the condition is the smallest possible convergence range. Consequently the Laplace integral exists only within some part of the complex plane. This part is called convergence area, as shown in Figure A.1. The variable is called
the abscissa of convergence. For values of with Eq. (A.1) makes no sense. Thus for the limit value of for must go to zero, but not for .

Example A.1.1

The Laplace integral converges for all with , as for decreases faster than any power of increases.

Next: The inverse Laplace transform Up: The Laplace transform Previous: The Laplace transform   Contents
Christian Schmid 2005-05-09