I am watching this video on complex Fourier Series where the instructor states the formula as:
$$ f(x) = C_0 + \sum_{-\infty}^{\infty}C_ne^{inx} $$

where as the notes on the same topic by Cambridge Uni state the formula as:
$$ f(x) = \sum_{-\infty}^{\infty}C_ne^{inx} $$

Which is the right formula?

  • $\begingroup$ Presumably in the first formula summation is taken over $n\in\mathbb{Z}\setminus\{0\}.$ Then both of these formulas are equivalent. $\endgroup$ – M. Strochyk Jul 14 '13 at 15:29
  • $\begingroup$ The second since... it's nicer ! :-) $\endgroup$ – Raymond Manzoni Jul 14 '13 at 15:50
  • $\begingroup$ Don't you sense a redundancy in the first sum? It's not that it's wrong, it's just...redundant. $\endgroup$ – Ron Gordon Jul 14 '13 at 16:51
  • $\begingroup$ @RonGordon I figured it out later that they put a condition in the second case that $n$ not equals 0 :) $\endgroup$ – Little Child Jul 14 '13 at 17:59
  • 2
    $\begingroup$ Then what M Strochyk said. $\endgroup$ – Ron Gordon Jul 14 '13 at 18:20

Both formulas say the same thing, since the first one should be written more precisely as $$f(x) = C_0 + \sum_{n\in\mathbb Z\setminus\{0\}} C_ne^{inx}$$ With complex Fourier series, there is little reason to separate the $0$th term from the rest of the sum. (One situation when it's done is when you want to write down $\int f(x)\,dx$.)

With trigonometric series it's very common to separate the $0$th term: $$f(x)=A_0+\sum_{n=1}^\infty (A_n\cos nx+B_n\sin nx)$$ because including $B_0\sin 0x$ feels silly.


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