**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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Tags: Convergence, Exponential Order, Laplace Transform, Math, Monday Math

This entry was posted on December 14, 2009 at 1:13 am and is filed under Math/Science. You can follow any responses to this entry through the RSS 2.0 feed.
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December 17, 2009 at 12:27 pm |

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!

March 1, 2010 at 12:08 am |

[…] 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 […]