Echoing “Fourier Echoes Euler”


The above tweet is from the Analysis Fact of The Day feed, which for the 5th had a neat little bit taken from Joseph Fourier’s The Analytic Theory Of Heat, published 1822. Fourier was trying to at least describe the way heat moves through objects, and along the way he developed thing called Fourier series and a field called Fourier Analysis. In this we treat functions — even ones we don’t yet know — as sinusoidal waves, overlapping and interfering with and reinforcing one another.

If we have infinitely many of these waves we can approximate … well, not every function, but surprisingly close to all the functions that might represent real-world affairs, and surprisingly near all the functions we’re interested in anyway. The advantage of representing functions as sums of sinusoidal waves is that sinusoidal waves are very easy to differentiate and integrate, and to add together those differentials and integrals, and that means we can turn problems that are extremely hard into problems that may be longer, but are made up of much easier parts. Since usually it’s better to do something that’s got many easy steps than it is to do something with a few hard ones, Fourier series and Fourier analysis are some of the things you get to know well as you become a mathematician.

The “Fourier Echoes Euler” page linked here shows simply one nice, sweet result that Fourier proved in that major work. It demonstrates what you get if, for absolutely any real number x, you add together \cos\left(x\right) - \frac12 \cos\left(2x\right) + \frac13 \cos\left(3x\right) - \frac14 \cos\left(4x\right) + \frac15 \cos\left(5x\right) - \cdots et cetera. There’s one step in it — “integration by parts” — that you’ll have to remember from freshman calculus, or maybe I’ll get around to explaining that someday, but I would expect most folks reading this far could follow this neat result.

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Author: Joseph Nebus

I was born 198 years to the day after Johnny Appleseed. The differences between us do not end there.

5 thoughts on “Echoing “Fourier Echoes Euler””

  1. Great coincidence – I have just discovered that Excel can do Fourier transforms. I tried to find some hidden periodicities in the daily average ambient temperature, but the FFT has just a single peak, corresponding to 365 days :-)

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