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### 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.

the Laplace transform.

- Doctor Roy, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
College Calculus
High School Calculus

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