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Sometimes when deriving the formulas for the coefficients of Fourier series mathematicians start with this definition:

$$f(t):=a_0+\sum_{n=1}^{\infty}\left[a_n\cos\frac{n\pi t}{L}+b_n\sin\frac{n\pi t}{L}\right]$$

But other times they start with:

$$f(t):=a_0+\sum_{n=1}^{\infty}\left[a_n\cos nt+b_n\sin nt\right]$$

The second one seems more intuitive but what's the intuition behind the first one? Are they equivalent?

mrk
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1 Answers1

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The only thing that differs is the domain (or the period). In the former case, $f:[-L,L]\to\mathbb{C}$ while in the latter case, $f:[-\pi,\pi]\to\mathbb{C}$.

They are practically equivalent. One can easily transform one case into the other by scaling $f$ in the $x$-direction.

md2perpe
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    @mrk. I realized that the domains were wrong. But your question probably remains. The answer is that the given domains cover one period of the functions. You can see that $f(t+2\pi) = f(t)$ in the latter case and $f(t+2L) = f(t)$ in the former case. – md2perpe Sep 08 '20 at 18:09