Integral over 2 pi of Cosine of n x/Mistake

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Mistake

$\ds \int_0^{2 \pi} \cos n x \rd x = 0$


It is implicit that $n$ ranges over all integers.

The author has neglected to specify the case where $n = 0$:

$\ds \int_0^{2 \pi} \cos n x \rd x = \begin {cases} 0 & : n \ne 0 \\ 2 \pi & : n = 0 \end {cases}$

As it stands, the Fourier coefficient $a_0$ is glossed over.


Sources