

Power Series Applications
Name: Michael B.
Status: student
Age: N/A
Location: N/A
Country: N/A
Date: N/A
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 HNMR (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 reexpress 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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Update: June 2012

