FrediFizzx wrote:gill1109 wrote:FrediFizzx wrote:FrediFizzx wrote:Gull's number (4) is correct but he neglected to mention that those 3 non-zero FT components are Dirac Delta infinite spikes.
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According to what he had to use for number (3) to make it correct, one of the two is wrong. If you integrate a cyclical function from -infinity to +infinity of course you will get infinity. It is silly to do that transform for a cyclical function. We are typically only interested in what happens over one cycle.
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Exactly. You need the Fourier series, not the Fourier transform. ...
Nonsense.
The Fourier Series of that function is just the same as converting the function to exponentials.
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Yes, exactly!!!!
The "Fourier series" is, strictly speaking, a *sequence* of complex numbers indexed by n in Z, found by applying the discrete Fourier transform. Your sequence has just three non-zero coefficients.
The reverse discrete Fourier transform takes the sequence, multiplies those numbers by exp(i n theta), and sums over n. That infinite summation is what Mathematica is showing you. In your case, it is a finite summation of just three terms.
When you ask Mathematica for the Fourier series of a function, it simply rewrites your function as sum c_n exp(i n theta).
The terminology is a bit tricky, and not everyone uses the same.
Can you ask Mathematica for the sequence of Fourier coefficients?