Monday Math 101

The Laplace Transform
Part 2: Convergence

When considering the Laplace transform
,
we must consider the region of convergence; those values of the complex number s for which the above integral converges.

A function f(t) is of exponential order at infinity if and only if there exist real numbers M and a, and positive real number T such that for all t>T; note then that
for .
Thus, we see that if the function f(t), t≥0, is piecewise continuous and has exponential order at infinity with , then the Laplace transform is defined for .
In fact, it can be shown that if the Laplace transform converges for some , then it converges for any s with , and thus the region of convergence is (if not the entire complex plane) a half-plane of the form , with all or some of the boundary possibly included.

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2 Responses to “Monday Math 101”

  1. The skepTick Says:

    Do you think you’ll do a post on Chaitin numbers or Benford’s law? I’m not sure if these curiosities fall within the thrust of your blog, but I sure wouldn’t want to miss it if you did.

    Always interesting reading. Thanks!

  2. Monday Math 110 « Twisted One 151's Weblog Says:

    […] Laplace transform F(s); thus the integral makes a more reasonable antiderivative of F(s) over the region of convergence. Note that if we define , then G’(s)=-F(s), and G(∞)=0 (as needed for it to be the […]

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