Integral over 2 pi of Cosine of n x/Mistake

Source Work

 * : Chapter One: $\S 2$. Fourier Series: $(4)$

Mistake

 * $\displaystyle \int_0^{2 \pi} \cos n x \, \mathrm d x = 0$

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

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


 * $\displaystyle \int_0^{2 \pi} \cos n x \, \mathrm d x = \begin{cases} 0 & : n \ne 0 \\ 2 \pi & : n = 0 \end{cases}$

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