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

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

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