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.


Tags: , , , ,

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 […]

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Google+ photo

You are commenting using your Google+ account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )


Connecting to %s

%d bloggers like this: