Complex Conjugate (FourierSeries)

Question

Let $f$ be a piecewise continuous complex function on the interval $[\pi,\pi]$ and \begin{equation} f(x) \sim \sum_{n=-\infty}^{\infty}c_{n}e^{inx} \tag{*} \end{equation} be its complex Fourier series. What is its relationship to the Fourier series of $f(\overline{x}),\overline{f(x)}$, and $f(-x)$?

Answer 1

$$\overline{f(x)}=\sum_{n\in\mathbb Z} \bar{c}_n e^{-inx}= \sum_{n\in\mathbb Z} c_{-n} e^{-inx}= f(x),$$

as we sum over $\mathbb Z$. Moreover

$$f(\bar{x})=\sum_{n\in\mathbb Z} c_n e^{in\bar{x}}=f(x),$$

as $x$ is real.

In general $$f(-x)=\sum_{n\in\mathbb Z} c_n e^{-inx};$$

if $f$ is a periodic even function, then $f(-x)=f(x)$, if odd $-f(-x)=f(x)$.