An exercise from Stein and Shakarchi's "Fourier Analysis" (exercise 14, chapter 8)

I'm trying to solve Exercise 14 of Chapter 8 of Fourier Analysis by Stein and Shakarchi. The problem is as follows:

The series $$\sum_{\vert n\vert\ne 0}\frac{e^{in\theta}}{n},\quad \mbox{for}\ \vert\theta\vert <\pi$$
converges for every $\theta$ and is the Fourier series of the function defined on $[-\pi,\pi]$ by $F(0)=0$ and
$$F(\theta) =
\begin{cases}
i(-\pi-\theta), & \text{if $-\pi\le \theta<0$} \\

i(\pi-\theta), & \text{if $0< \theta\le \pi$}
\end{cases}$$
and extended by periodicity (period $2\pi$) to all of $\mathbb R$

Show also that if $\theta\ne 0 \mod 2\pi$, then the series

$$E(\theta)=\sum_{n=1}^{\infty}\frac{e^{in\theta}}{n}$$
converges, and that
$$E(\theta)={1\over2}\log\left({1\over{2-2\cos \theta}}\right)+{i\over 2}F(\theta)$$
And I do not know how to prove the last identity. Are there any hints?