Finding real part of fourier series

Question

I have encountered the following problem in one of my textbooks but I'm not really getting anywhere:

Let $f$ be complex-valued and piecewise continuous on the interval $[-\pi,\pi]$. Find the complex fourier series of $Re(f)$ on the basis of the complex Fourier series of $f$.

i.e we're given that $f$ has the complex Fourier coefficient $c_n$ and we want to express $Re(f)$'s fourier coefficients, $C_n$, in terms of $c_n$. Did i interpret this correct?

I've tried splitting $f=g+ih$ but i'm not managing to express $C_n$ in terms of known objects.

Thanks in advance.

Answer 1

$\Re(f) = \frac{1}{2}(f + \overline{f})$, so $C_n = \hat{\Re(f)} = \frac{1}{2}(\hat{f}+\hat{\overline{f}}) = \frac{1}{2}(c_n + \overline{c_n}).$ Is that what you'd like?

Answer 2

Hint
$$e^{inx} = \cos(nx) + i\sin(nx)$$ Look an $c_n$ and $c_{-n}$ at the same time.