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One of the deepest reasons to use the Fourier expansion is the idea of Harmonic Analysis induced by symmetries. The main topic is called Representation Theory and it forms the foundation to large parts of physics. Here's the idea: consider a (finite - or compact, for which many similar properties hold) group $G$ and a vector space $V$ on which it operates. ...


3

Many of the other answers are addressing the practicalities of expanding in Fourier series versus Taylor series. But there is at least one physical reason for choosing one over the other, and that is that the expansion coefficients of a vector written in an orthonormal basis reveal particular types of physical information about the system being described by ...


2

$T=1.94$ sec is certainly compatible with the big, but noisy peak at .5Hz. Why are you worrying about the tiny peak at about 2.6Hz? Your data looks very noisy so random but meaningless peaks are to be expected. I still don't understand why your FFT is so noisy given the smooth data of black points in the first plot. You say that you interpolate? Why do ...


2

For a down to earth but rigorous account distributions and delta functions (but not so much differential equations) you can beat James Lighthill's Introduction to Fourier analysis and generalised functions, Cambridge University Press. ISBN 978-0-521-05556-7. The book is quite thin, only 70 pages or so. It is written at the undergraduate level. Although ...


1

You may want to check out the book by Matthews and Walker. This is the book I used for "mathematical methods" class. If I remember correctly, it's based on lectures by Feynman. It's very concise, although I don't know that it has enough "real world" examples to satisfy your taste.


1

Trying to replicate the steps in the text the first they do is say that $\int d^4x (\partial_{\mu}\phi(x_i))^2-m^2\phi(x_i)=\frac{1}{V}\sum_n(|k_n|^2-m^2)|\phi(k_n)|^2$, simply writing the function as a fourier series should be enough but even so at first I wasn't able until it clicked that this equality is the plancherel/parseval identity for derivatives ...


1

Fourier series let you model your space of potential solutions as an infinite dimensional analogue of a finite dimensional Euclidean vector space. All of your formal analytical arguments can take place in that infinite dimensional space, where you can use your Euclidean intuition. For example, in place of an inner product $\langle u,v\rangle = \sum_{i=1}^n ...


1

From the perspective of complex numbers, Fourier series are polynomials — or at least, Taylor series. For example, take the real period $1$ function $f$ with $f(x) = x$ for $-1/2 \le x < 1/2$. The Fourier series of $f$ is $$f(x) = \sum_{n=1}^\infty \frac{(-1)^{n-1}}{\pi n} \sin 2\pi n x.$$ Using complex numbers, this can be proved as follows: for $...


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