Laplace transform interpretation

[leehufford]

Hello,

We were introduced to Laplace transforms in my ODE class a few days ago, so naturally I went online to try to figure out what this transform actually is, rather than being satisfied with being able to compute simple Laplace transforms.

The Wikipedia page for the transform says it takes the input from the "time domain" into the "frequency domain" where the argument is now a complex angular frequency in radians per unit time.

The page also says the Laplace transform resolves a function into its moments.

If someone could use the very simple example of the Laplace transform of 1 becoming 1/s to explain what is meant by both the time domain to complex frequency domain transform and also to explain what it means to resolve a function into into its moments, It would absolutely make my day. Thank you so much for your time,

-Lee

 

[the_wolfman]

Hey Lee,

The Laplace transform of 1 isn't going to illustrate this point.
A better example is the Laplace transform of a sine wave $\sin \omega t$ or better yet $a_1 \sin \omega_1 t+ a_2 \sin \omega_2 t + \dots$

Plot the amplitude of the transform in the complex plane (or just along the imaginary axis). What happens at $S = i \omega$ ?

Hopefully this helps.

 

[AlephZero]

Moment generating functions are mostly used in probability theory. This http://www.pitt.edu/~super7/19011-20001/19461.pdf defines what they are and how they relate to Laplace transforms. It doesn't give much motivation, but if you had studied enough statistics to understand the motivation, you would probably have seen the connection without asking the question.