S.O.S. Mathematics CyberBoard

[jddipqd]

I need to find the inverse Laplace of the following function:

[Max Power]

One way to do this is to use the well known Bromwich Integral for the Inverse Laplace transformation, but it demands some work to be done:

Where is chosen so that every singularity of
is on the left hand side of the line .
Then if is a meromorphic function (analytic everywhere but in a finite set of points and with no essential singularities) on the left hand side of that line (on the complex plane), then you can calculate this integral (relatively) easily using contour integration (ie. extend the line to an infinite semicircle on the left (since e^zt has an essential singularity on the right) and voilá, you have solved the problem.

In order to compute the contour integral you have to find the singularities (in the case of a meromorphic function they are poles of finite order) and calculate their residues.

A complex function has an m-order pole at iff.

So for example has a first order (or simple) pole at .

Then if a complex function has a m-order pole at then the residue at that point is:


And the theory of contour integration states that
where B is an area in the complex plane is its boundary, nRes = the number of poles within B and is a finite order pole.

Now your function
has two third order poles at (or one third order at the origin when ) so the contour integration gives

and of course

And the rest is up to you. (I Don't have the time to compute the derivatives since there is quite a lot of work in it, but you should be ok)
If you still have troubles then ask away and the proofs for the statements I made above can be found in any advanced book on complex calculus.