Broken image methinks.

I haven't actually studied Laplacian Transforms yet, but from what I've seen, you basically take a Taylor series and generalise it to a continuous case. Obviously you can't explicitly give the coefficients any more if there are infinite number of them, so they're written as a function - this is the function that you 'take the transform' of, and it gives you the new function.

It's pretty straightforward working out what the transform should be with this in mind, and then it's also easy to see that this is equivalent to the way that it's normally written (it's in a slightly weird form because it turns out that calculations are just easier when it's written the way that it is).