Laplace TransformDate: 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/ |
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