The Inverse Laplace Transform

I'm nearing the end of Math 20D: Differential Equations. It's an introductory course, and in fact, the last lecture was earlier today and the final is a week from today. In the last week, we were taught how to do the forward Laplace transformation. However, to invert the Laplace transformations, we were told to compare the result of our findings to a table of already known forms of Laplace transformations.

I'm probably not going to take any higher math, because higher math does not fit into my major requirements.

However, I feel like only going forward and not knowing how to go backwards is unsatisfying. Knowing only one half of the puzzle is incomplete, not general, and makes the whole process somewhat useless and definitely meaningless. I already know how to solve most of the forms of differential equations (up to second order systems) that I've already seen without the Laplace transform. Therefore, the Laplace transform doesn't reveal anything new to me. It really just seems pointless.

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

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?

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.

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