Laplace Transform

Date: 01/25/2001 at 07:16:34
From: ronald
Subject: Laplace transform

I can not visualise what Laplace transforms are all about. Can you 
send me some simple notes? What are their applications in real life ?

Date: 01/25/2001 at 13:35:32
From: Doctor Roy
Subject: Re: Laplace transform

Hello,

Thanks for writing to Dr. Math.

The Laplace transform is a simple way of converting functions in one 
domain to functions of another domain. Here's an example:

Suppose we have a function of time, such as cos(w*t). With the Laplace 
transform, we can convert this to a function of frequency, which is

    cos(w*t)  ----L{}----->  w / (s^2 + w^2)

This is useful for a very simple reason: it makes solving differential 
equations much easier. Let me demonstrate by analogy. Several hundred 
years ago, logarithms were developed that aided astronomers 
tremendously. This is because a logarithm provided a simple way to 
transform difficult multiplications into addition, by the following 
logarithm rule:

      log(a*b) = log(a) + log(b)

Although this may seem simple, the development of the logarithm was 
considered the most important development in studying astronomy.

In much the same way, the Laplace transform makes it much easier to 
solve differential equations. Since the Laplace transform of a 
derivative becomes a multiple of the domain variable, the Laplace 
transform turns a complicated n-th order differential equation to a 
corresponding nth degree polynomial. Since polynomials are much easier 
to solve, we would rather deal with them. This occurs all the time. 

For instance, consider any type of communications system: FM/AM 
stereo, cellular phones, 2-way radios, etc. Sending signals over media 
in these systems is much simplified by considering the signals by 
their frequency content instead of their actual time functions.  This 
is simply a Laplace transform.  

Consider also any circuit board. A circuit board can be modeled by a 
horrendous system of differential equations. These are not fun to 
solve. However, by using the Laplace transform, we can solve for any 
parameter of a circuit by solving the appropriate polynomial. 

In brief, the Laplace transform is really just a shortcut for complex 
calculations. It may seem troublesome, but it bypasses some of the 
most difficult mathematics.

I hope this helps. Please write back if you want more information on 
the Laplace transform.

- Doctor Roy, The Math Forum
  http://mathforum.org/dr.math/