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: 
The Newton iteration is
$$
x\_{n+1} = x\_n - \frac{f(x\_n)}{f'(x\_n)}
$$
and to find the inverse you should root-find $f(x) = 1/x - R$ which has a root at $1/R$. Note that $f'(x) = -1/x^2$ so $f(x\_n)/f'(x\_n)$ will give you exactly the iteration you are looking for.
The function $f(x) = R - \frac{1}{x}. \;$ You find the simple derivation of your recursion formula [here](http://en.wikipedia.org/wiki/Newton%27s_method#Multiplicative_inverse_of_numbers_and_power_series)