Given a function $f \in L^2(-\pi,\pi)$ Parseval's theorem states $$ \frac{1}{2\pi} \int_{-\pi}^{\pi}|f(x)|^2 dx=\sum_{n=-\infty}^{\infty}|c_n|^2$$
Is the following also true? $$ \frac{1}{2\pi} \int_{-\pi}^{\pi}|f(x)|^2 dx=2\sum_{n=1}^{\infty}|c_n|^2 +c_0$$
I was thinking it would be true because we have $$c_{-n}=\bar{c_n}$$ and $$|c_n|=|\bar{c_n}|$$
Is that valid?
You are right if $f$ is a real function. But if $f$ takes complex non-real values this is not true anymore. Note that, in this case, it is not true that$$(\forall n\in\mathbb{Z}):c_{-n}=\overline{c_n}.$$