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I've recently started learning about Fourier series from this set of tutorials: (http://lpsa.swarthmore.edu/Fourier/Series/WhyFS.html) and a few other sources.

While chugging through the material, I noticed and was intrigued by the similarity of the derivation of the coefficients for the even Fourier series, odd Fourier series, and exponential Fourier series. I'll explain what I noticed in the following two paragraphs.

The first step in all three derivations I looked at is to multiply both sides of the equation x(t)=sum(etc...) by the non-coefficient term inside of the sum. For the even and odd Fourier series, this is a cosine function. For the exponential Fourier series, this is a natural exponential function. The derivation also requires replacing the index value inside of that function with a new independent variable.

The next step in all three derivations is to integrate both sides of the equation over one period. From here, the side of the equation with the sum works out to be the coefficient multiplied by the period in all three derivations, which one then divides by the period to isolate the coefficient.

There are probably more similarities between these derivations, and it makes sense that they would be similar, since they are all Fourier series. I was wondering, though, if there is some concrete mathematical reason that all these derivations, particularly the exponential vs. the other two, end up with lots of similarities. Is there something deep, profound, and interesting going on here?

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Yes, there is something profound. Spaces of real and complex valued functions can be viewed as infinite dimensional vector spaces since they can be added and multiplied by numbers. In some of these spaces you can introduce scalar product (which allows you to introduce distances and angles). What you are doing when you calculate particular coefficient in your expansion is finding projection of function on given subspace - just as you would calculate projection of a vector in Euclidean space onto given axis. Each vector can be described by giving its components in some system of "axis" and the same can be done with functions. Difference is that in this case scalar product is given by integral. If you want to learn something more about it, look up "Hilbert spaces".

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