The use of Laplace Transforms in control theory and in signal analysis took off after WWII and has become a widely established tool of analysis. When I was a student in the dark ages (before the Internet and reality TV), we used look up tables to determine Laplace Transforms and their inverse. These tables are still widely used but online tools like Wolfram Alpha can also be readily used.

Like many aspects of Wolfram Alpha, the commands are largely intuitive and the program is forgiving with syntax:

For example, type in "Laplace Transform (t sin(2t))" into the dialogue box will produce the following answer and a graph of the function F(s). The graph of the transform is a nice bonus above the traditional look up tables, as it helps appreciate the new function that you formed through this transformation.

The inverse operations are just as simple. For example, the program only blinked for a few moments on "Inverse Laplace Transform (3s /(s^2 + 6))" to produce "3 cos(SQRT(6) t)". Once again a useful graph is generated and can be downloaded.

I encourage you to test how good Wolfram Alpha is at solving the inverse problems. Man (and Woman) against machine is always fun.

I'm sure Marquis de Laplace would have been impressed !!

The inverse operations are just as simple. For example, the program only blinked for a few moments on "Inverse Laplace Transform (3s /(s^2 + 6))" to produce "3 cos(SQRT(6) t)". Once again a useful graph is generated and can be downloaded.

I encourage you to test how good Wolfram Alpha is at solving the inverse problems. Man (and Woman) against machine is always fun.

I'm sure Marquis de Laplace would have been impressed !!