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?
Best Answer
Observe that
$$\sum_{n=1}^\infty\frac{e^{in\theta}}n=\sum_{n=1}^\infty\frac{\cos n\theta}n+i\sum_{n=1}^\infty\frac{\sin n\theta}n$$
Splitting the sum above is justified because both series above converge, for example using Dirichlet's test (read Showing $\sum\frac{\sin(nx)}{n}$ converges pointwise and also http://mathforum.org/library/drmath/view/72101.html , for instance) , and this already proves convergence for $\;\theta\neq2k\pi\;,\;\;k\in\Bbb Z$ , since for $\;\theta=2k\pi\;$ we get the harmonic series.
Now, since for $\;z\in\Bbb C\;,\;\;|z|<1\;$ we have:
$$\frac1{1-z}=\sum_{n=1}^\infty x^{n-1}\implies -\text{Log}\,(1-z)=\sum_{n=1}^\infty\frac{z^n}n\;+\;K\text{ (=constant)}\implies$$
(Log$\,\,z\,$ is the complex logarithm) substitute $\;z=e^{i\theta}\;$ (this is justified by Abel's Theorem) :
$$-\text{Log}\,(1-e^{i\theta})=\text{Log}\,\frac1{1-e^{i\theta}}=\sum_{n=1}^\infty\frac{e^{in\theta}}n\;+\;K$$
Finally (fill in details of all the above), observe that
$$1-e^{i\theta}=1-\cos\theta-i\sin\theta\implies |1-e^{i\theta}|=\sqrt{2(1-\cos\theta)}$$ and also
$$\text{Log}\,z:=\log|z|+i\arg z\;\;,\;\;\text{with}\;\;\log\;\;\text{the real usual logarithm}$$
and usually choosing the main branch's principal value for the logarithm, in which $\;\arg z\in(-\pi,\,\pi]\;$ .
The summand $\;\frac i2F(\theta)\;$ is the constant above