Laplace Transforms, Region of Convergence

[tanky322]

Can anyone explain the region of convergence to me in english? I understand the Laplace transform and can do it with my eyes closed, but I can't figure out how to figure out the ROC. I've scoured the internet, and every definition is vague or just incomprehensible by me.


Thanks!

 

[lurflurf]

Do you know anything about complex variables? The region of convergence is just the values of t where
\int_0^\infty f(t)e^{-s t} dt
converges as an improper integral.
That can be difficult to find in general, but in many elementary applications only very well behaved f are considered for example the functions of exponential order.

 

[tanky322]

I'm somewhat familiar with complex variables, although not too much. I guess what I am not really sure of, is what exactly converges? The function and e^-st?


Thanks for your reply!

 

[g_edgar]

What exactly converges? The improper integral. That is, the limit
\lim_{M\to+\infty}\int_0^M f(t)e^{-s t} dt
exists. Generally, the region of convergence is a half-plane: all s to the right of some vertical line in the complex plane.

 

[jasonRF]

As an example to what g_edgar wrote, consider
f(t) = e^{5 t},
for
t\geq0.

Now calculuate

<br /> \lim_{M\to+\infty}\int_0^M f(t)e^{-s t} dt = \lim_{M\to+\infty}\int_0^M e^{-(s-5) t}.<br />

You should find that the limit only converges if s satisfies some condition. That condition defines the region of convergence. Note that s is complex in general, and the constraint will be on the real part.

Jason