\begin{align} f(x)&=\sum_{n=-\infty}^{\infty}c_ne^{inx}\\c_n&=\displaystyle\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-inx}dx\\ \end{align}

\begin{align} f(x)&=\sum_{n=-\infty}^{\infty}c_ne^{inx}=\sum_{n'=-\infty}^{\infty}c_{-n'}e^{-in'x}\\ f(x)^\ast&=\sum_{n=-\infty}^{\infty}c_n^\ast e^{-inx}\\ \end{align}