The Inverse Laplace Transform

Thus, my first question is: What is the point of the Laplace Transformation?

The idea behind the Laplace Transformation is that you turn a differential equation into an algebraic equation. By solving the algebraic equation, you reverse the Laplace Transformation and get the specific solution back in. In many cases, this is way faster than solving the actual differential equation.

My second question is: If the Laplace transform is a function of s, why can we just move e^-s in and out of the integral as though s were a constant? I know this is because exp(-s*inf)=0, but this still doesn't justify moving s in and out of the integral. In fact, s is really treated as a constant throughout the entire process. Even when we integrate exp(-st), we get -exp(st)/s. So what is the meaning of emphasizing that the Laplace transform as a function of s? Why not just call it a function of t, and say that s is an arbitrary constant?

You cannot simply move e^(-s) out and in of the integral because you have e^(-st), not e^(-s). I've never seen that you move e^(-s) out of the integral.

My third question is: How do you use the following inversion formula?



What does it mean? Is there an example for this? I'm not interested in an exhaustive proof as much as I am interested in its application and the method of its use. Keep in mind that I already scourged Google, and all I ended up with was a bunch of websites telling me to convert by comparing to previously known Laplace transforms.

You don't usually use the inversion formula, you just use previously existing techniques to solve the Laplace Transformation. However, it is nice to know because it is used frequently in Fourier transformations by setting gamma equal to 0.

Noah