Laplace Transform

According to the wiki, “Laplace Transform” transforms a function of a real variable t (often time) to a function of a complex variables (complex frequency). The transform has many applications in science and engineering.

It’s similar to Fourier transform with a difference that Fourier transform of a function is a complex function of a real variable, while the Laplace transform of a function is a complex function of a complex variable.

Then what is Laplace Transform, and of course, an alternate notation for the Laplace transform is 

{\mathcal {L}}\{f\}

Right from the beginning, we need to know why mathematicians such as Lagrange, Laplace, Fourier would come up with all sorts of “transforms”, simple, they just want to solve differential equations after Newton and Leibniz invented calculus. It’s not easy to solve these Differential Equations(DEs).

So Laplace come up with this Laplace Transforms, and painstakingly built up a table of Laplace equations. For example, to solve this DE:

y'(0) = 0 y(0) = 0

Established tables find (all proved) that

It’s eventually solved as (per Khan Academy)

Now let’s get into more hairy territory, note the bottom equation is the original form, while the top one is deduced thanks to the bridge aid in the middle.

And the crux is that s represents frequency domain while t is the time domain, sounds familiar to Fourier transform, right?

In the application of physics, wiki provides an example of how Laplace transform helps to solve the radioactivity decay equation.

If we can solve this DE to get a normal function to time t, so we can predict what’s the radioactive metric after certain time.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

This site uses Akismet to reduce spam. Learn how your comment data is processed.