One of the marvels of mathematics is the Fourier series which shows how to either decompse a complex function into a sum of sines and cosines or vice versa build up a complex function by summing over various sines and cosine functions. This idea has many applications in partial differential equations and any function expressed in rectangular co-ordinates can be expressed in terms of them. The separate sine and cosine terms can be seen as vectors in a function space. In that they are orthogonal to each other thus there is a an amazing analogy between geometric vectors and vectors in function space. This can be extended to other coordinate systems as we shall see in the next two poems. One of the most amazing application of Fourier series is the ability to synthesize complex sounds and make new music. An application that I bet Fourier never thought of. So next time you are struggling to solve a problem in Fourier series just remember how useful they can be. Here is the tribute.
Mr
Fourier
You
discovered something quite profound,
How
to create a complex sound.
From
it’s component bits,
To
make a wave that fits.
All
these waves combine,
Into
a sound that’s fine.
The
possibilities are endless,
To
create sounds quite stupendous.
A Moog synthesiser must be much more fun playing around with one of these than the digital synthesisers which you can generate using a computer A concrecte application of Fourier series.
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