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Name: Michael B.
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Question:
We emphasize applications throughout our Calculus course. For Power Series, however, I know little of how they are applied in physics, engineering, and other fields. Can you cite some applications or direct me where to look?



Replies:
Michael,

1.) FTIR spectrsocopy (Fourier Transform infrared spectroscopy).

I believe uses a progressively recurring algorithm (which incorporates power series expansions) that slowly converge on a solution. Who cares, you might say. Organic chemists use this method of spectroscopy to identify molecules.

2.) C13+ and H-NMR (carbon - 13 and proton NMR) both use recursive power series expansions algorithms that reconstruct a spectr4al fingerprint of an unkown molecules so that it may be compared with spectral fingerprints of known molecules for a possible match.

AND HERE iS MY FAVORITE.

3.) SPECTRUM ANALYZERS.

A spectrum analyzer will take as its input a "spectrum" of signals. For example, the output of a Motorola talk about radio. At work, I might use a Rhode Schwarz spectrum analyzer to view the magnitude vs frequency response of a Motorola radio...or any other for that matter. My point here is that a spectrum analyzer takes a, lets say a voltage signal, and "bins out" all of the singular frequency points within that sampling. And much thanks to the French mathematician, Fourier, we have developed a way to quickly calculate the Fourier constants to (in real time) construct a very accurate SPECTRAL response. THIS STUFF IS VERY VERY IMPORTANT. I cannot stress how important learning power series expansions are. As an engineer, of course, I would stress you to spend most of your time on Laplace transforms and Fourier. I believe that one is a special case of the other. But my math is a little rusty.

I hope this has helped.

Darin Wagner


You might want to check an excellent book on applied math: Mathematical Methods in the Physical Sciences by Mary L. Boas.

Rich Robinson


Power series are infinite series that contain terms X^r, that is X raised to a series of exponents. Their applications to applied fields are so ubiquitous it is difficult to know where to start. All the "trig" functions, log functions, calculating the payback on an amount of invested money, many differential equations, numbers like 'pi' and other transcendental numbers all have associated power series. There are some very elegant infinite series such as the power series expression called Fibonacci series (you can find a lot of books on these). There are special groups of power series such as Taylor series that find wide use in many applications. A standard "trick" physicists and engineers use is to re-express terms in some complex equation by their associated power series. Then they can solve the problem as a sum of individual powers term by term to any precision the problem requires. Hope this helps.

Vince Calder



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